GIFT  TO  THE  LIBRARY 

DEPARTMENT  OF  CIVIL  ENGINEERING 
UNIVERSITY  OF  CALIFORNIA 

FROM 

UNIVERSITY  OF  CALIFORNIA  STUDENT 

CHAPTER,  AMERICAN  SOCIETY 

OF  CIVIL  ENGINEERS 


UNIVERSITY   OF  CALIFORNIA 

DEPARTMENT  OF  CIVIL   ENGINEERING 

BERKELEY,  CALIFORNIA 


Engineering 
T  ;K».Q, 


UNIVERSITY   OF  CALIFORNIA 
=»ARTMENT  OF  CIVIL   ENGINEERING 
BERKELEY,  CALIFORNIA 


RETAINING  WALLS 

THEIR  DESIGN  AND  CONSTRUCTION 


%  Qraw-MlBook  (a  7ne 


PUBLISHERS     OF     BOOKS 

Coal  Age     v     Electric  Railway  Journal 

Electrical  World  •*  Engineering  News-Record 

American  Machinist  v  Ingenieria  Internacional 

Engineering  S  Mining  Journal    '  ^     Po  we  r 

Chemical  6    Metallurgical  Engineering 

Electrical  Merchandising 


EETAINING  WALLS 

THEIR   DESIGN 
AND  CONSTRUCTION 


BY 
GEORGE  PAASWELL,  C.  E. 


FIRST  EDITION 


McGRAW-HILL  BOOK  COMPANY,  INC. 

NEW  YORK:    239   WEST  39TH  STREET 

LONDON:    6  &  8  BOUVERIE  ST.,  E.  C.  4 

1920 


Engineering 
Library 

COPYBIGHT,    1920,   BY   THE 

MCGRAW-HILL  BOOK  COMPANY,  INC. 


THK  MAPLE  PRESS  YORK  PA 


PREFACE 


The  presentation  of  another  book  on  retaining  walls  is  made 
with  the  plea  that  it  is  essentially  a  text  on  the  design  and  con- 
struction of  retaining  walls.  The  usual  text  on  this  subject 
places  much  emphasis  upon  the  determination  of  the  lateral 
thrust  of  the  retained  earth;  the  design  and  construction  of  the 
wall  itself  is  subordinated  to  this  analysis.  Without  gainsaying 
the  importance  of  the  proper  analysis  of  the  action  of  earth 
masses,  it  is  felt  that  such  is  properly  of  secondary  importance  in 
comparison  with  the  design  of  the  wall  itself  and  the  study  of 
the  practical  problems  involved  in  its  construction. 

It  is  the  purpose  of  the  first  chapter  to  present  the  existing 
theories  of  lateral  earth  pressure  and  then  to  attempt  to  codify 
such  theories  evolving  a  simple,  yet  well-founded  expression  for 
the  thrust. 

An  attempt  is  made  to  continue  this  codification  throughout 
the  theories  of  retaining  wall  design  so  that  a  direct  and  continu- 
ous analysis  may  be  made  of  a  wall  from  the  preliminary  selection 
of  the  type  to  the  finished  section.  Such  mathematical  work  as 
is  presented  is  given  with  this  essential  object  in  view. 

Under  Construction  advantage  is  taken  of  a  classic  pamphlet 
on  Plant  issued  by  the  Ransome  Concrete  Plant  Co.  (which  pam- 
phlet should  be  in  the  possession  of  every  construction  engineer) 
to  illustrate  the  principles  of  proper  plant  selection. 

A  retaining  wall  is  a  structure  exposed  to  public  scrutiny  and 
must,  therefore,  present  a  pleasing,  but  not  necessarily  ornate 
appearance.  Since,  in  the  case  of  concrete  walls,  the  appearance 
of  the  wall  is  dependent  upon  the,  character  of  the  concrete  work, 
it  is.  essential  that  the  edicts  of  good  construction  be  observed. 
For  this  reason  the  modern  development  of  concreting  is  pre- 
sented fully  with  frequent  extracts  from  some  of  the  recent  im- 
portant reports  of  laboratory  investigators. 

It  is  hoped  that  proper  credit  has  been  given  to  the  authors  of 
all  such  quoted  passages,  as  well  as  to  other  references  used. 
A  vast  amount  of  literature  exists  on  the  subject  of  retaining  walls 

800493 


vi  PREFACE 

and  earth  pressure  (see  bibliography  at  the  end  of  the  book), 
and  in  view  of  the  absence  of  a  proper  collation  of  all  this  material 
there  is,  of  course,  much  duplication  of  the  analysis.  It  is  hoped 
that  before  future  studies  are  made  of  earth  pressure  phenomena, 
an  attempt  will  be  made  to  examine  existing  literature  and  that 
a  due  appreciation  will  be  had  of  the  subordinated  importance 
of  the  determination  of  lateral  pressure. 

I  must  take  this  opportunity  to  thank  Mr.  Arthur  E.  Clark, 
Member,  Am.  Soc.  C.  E.,  for  his  patient  reading  of  the  text  and 
his  many  helpful  hints. 

To  Mr.  F.  E.  Schmitt,  Associate  Editor  of  the  Engineering 
News-Record,  I  am  deeply  grateful  for  encouragement  and  aid 
in  preparing  the  book  and  in  arranging  the  subject  matter  in  a 
logical  and  clear  manner. 

THE  AUTHOR. 
NEW  YORK, 
Feb,  1919. 


CONTENTS 


PREFACE.    .  , v 

LIST  OF  PLATES vii 

PART  I 
DESIGN 

CHAPTER  I 

EARTH  PRESSURES 1 

History  of  the  various  theories  of  earth  pressure — Exact  analysis 
of  the  action  of  earth  masses — The  ideal  earth  and  the  fill  of 
actual  practice — The  two  theories — Rankine's  Theory — Cou- 
lomb's method  of  the  wedge  of  sliding — Various  other  methods  of 
thrust  calculations — Experimental  data — Wall  friction — Cohesion. 
— Surcharge — Pressure  on  cofferdams — Pressure  of  saturated  soils 
— Sea  walls — Problems. 

CHAPTER  II 

GRAVITY  WALLS 42 

Location  and  height  of  wall — General  outline  of  wall — Two  classes 
of  walls — Fundamental  principles  of  design — Concrete  or  stone 
walls — Thrust  and  stability  moments — Foundations — Distribution 
of  base  pressures — Factor  of  safety — Footing — Direct  method  of  de- 
signing the  wall  proper — Revetment  walls — Problems. 

CHAPTER  III 

REINFORCED  CONCRETE  WALLS 79 

General  principles — Preliminary  section — Distribution  of  base  pres- 
sures— Tables  and  their  use — Theory  of  the  action  of  reinforced 
concrete — Bending  and  anchoring  rods — Vertical  arm — Footing — 
Toe  extension — Counterfort  walls — Face  slab — Footing — Counter- 
fort— Rod  system — Problems. 

CHAPTER  IV 

MISCELLANEOUS  WALL  SECTIONS 122 

Cellular  walls — Hollow  cellular  walls — Timber  cribbing — Concrete 

vii 


viii  CONTENTS 

cribbing — Walls  with  land  ties — Walls  with  relieving  arches — 
Parallel  walls  enclosing  embankments — Abutments — Box  sections 
subject  to  earth  pressures — Advantage  of  the  various  types  of  walls 
— Problems. 

CHAPTER  V 

TEMPERATURE  AND  SHRINKAGE.     GENERAL  NOTES 151 

General  theory  of  the  flow  of  heat  and  the  range  of  temperature  in 
concrete  masses — Shrinkage — Settlement — Expansion  joints — 
Construction  joints — Wall  failures. 

PART  II 
CONSTRUCTION 

CHAPTER  VI 

PLANT 165 

Relation  between  plant  and  character  of  works — Standard  plant 
layouts — Sub-division  of  field  operations — Mixers — Distribution 
systems — Examples  of  plant  layouts. 

CHAPTER  VII 

FORM  WORK .181 

Sub-division  of  forms — Concrete  pressures — Major  Shunk's  experi- 
ments— Robinson's  experiments — Lagging,  joists  and  rangers — Tie 
rods — Bracing — Stripping  forms — Oiling  and  wetting  forms — Pat- 
ent Forms — Hydraulic,  Blaw — Supporting  the  rod  reinforcement 
— Examples  of  form  work — Problems. 

CHAPTER  VIII 

CONCRETE  CONSTRUCTION . 197 

Modern  developments — PROF.  TALBOT'S  notes  on  concrete — 
Conclusions  of  Bureau  of  Standards — PROF.  ABRAM'S  analysis  of 
concrete  action — Importance  of  the  water  content — PROF.  ABRAM'S 
conclusions — Application  of  theory  to  practice — Concrete  methods 
distributing  concrete — Keying  lifts — Use  of  cyclopean  masonry — 
Winter  concreting — Acceleration  of  concrete  hardening — Concrete 
materials — Cements — Sand — Crushed  stone  and  gravel — Fineness 
modulus — Method  of  surface  areas  (CAPT.  E.  N.  EDWARDS) — 
CRUM'S  method  of  proportioning  aggregates. 

CHAPTER  IX 

WALLS  OTHER  THAN  CONCRETE 227 

Plant  required — Mortar — Construction  of  wall — Coping — Face 
finish — Special  stone — Plaster  coats — Cost  data. 


CONTENTS  ix 

CHAPTER  X 

ARCHITECTURAL  TREATMENT;  DRAINAGE;  WATERPROOFING 232 

Architectural  treatment — Face  treatment — Rubbing — Tooling — 
Special  finishes — Colored  aggregates — Artistic  treatment  in  general 
— Hand  rails  and  parapet  walls — Drainage — Examples  in  practice 
— Waterproofing. 

CHAPTER  XI 

FIELD  AND  OFFICE  WORK.     COST  DATA 242 

Surveys  necessary — Construction  lines — Walls  on  curves — Lines 
for  concrete  forms — Computation  of  volumes — Isometric  repre- 
sentation of  wall  details — Cost  data — Labor  costs — Examples  of 
cost  of  work. 

APPENDIX 

SPECIFICATIONS;  BIBLIOGRAPHY;  GENERAL  INDEX 254 

INDEX.  .    271 


LIST  OF  PLATES 

FACING  PAGE 

PLATE  I 46 

Fig.  A. — Dry  rubble  wall  along  highway. 

Fig.  B. — Characteristic  appearance  of  cement  rubble  wall. 

PLATE  II 150 

Fig.  A. — Crack  in  reinforced  concrete  wall  at  junction  of  wing  wall 

and  abutment. 
Fig.  B. — Structural  steel  supports  for  special  type  of  retaining  wall. 

PLATE  III 150 

Crack  in  sharp  corner  of  wall  due  to  tension  component  of  thrust. 

PLATE  IV 191 

Fig.  A. — Unsatisfactory  rod  detail  for  concrete  pouring. 

Fig.  B. — Holding  vertical  rods  in  place  before  concrete  is  poured. 

PLATE  V 227 

Fig.  A. — Method  of  laying  stone  wall  by  series  of  derricks. 

PLATE  VI 227 

Fig.  A. — Uncoursed  rubble  wall  with  coursed  effect  given  by  false 

pointing. 
Fig.  B. — Rubble  wall  (Los  Angeles)  with  face  formed  by  nigger-heads. 

PLATE  VII 236 

Fig.  A. — Showing  effects  of  poor  concrete  work. 

Fig.  B. — Ornamental  parapet  wall.     Tooled  with  rubbed  border. 

PLATE  VIII 236 

Fig.  A. — Ornamental  handrail — approach  to  viaduct. 
Fig.  B. — Picket  fence  wall  lining  open  cut  approach  to  depressed 
street  crossing. 

PLATE  IX 236 

Fig.  A. — Ornamental  concrete  handrail  approach  to  concrete  arch. 


RETAINING  WALLS 

THEIR  DESIGN  AND 
CONSTRUCTION 


PART  I 
DESIGN 


CHAPTER  I 


THEORY  OF  EARTH  PRESSURE 


The  Development  of  the  Theory  of  Earth  Pressure.1 — A  search 
through  engineering  and  other  scientific  archives  fails  to  yield 
any  evidence  that  prior  to  1687  an  attempt  had  been  made  to 


B 


analyze  the  action  of  earth 

pressure    upon    a    retaining 

wall.      Undoubtedly,    rough 

methods   of   computing   wall 

dimensions   existed    back    in 

prehistoric    times,    since    the 

art  of  constructing  retaining 

walls  is  as  old  as  building  art 

itself.        In    1687    GENERAL 

VAUBAN,2  a  French  military 

engineer  gave  some  rules  for 

figuring  walls,  but  presented 

no  theoretical  basis  for  these 

rules.    It  is  questionable  whether  such  existed.    In  1691  BULLETS 

advanced  a  rather  primitive  method,  assuming  that  the  angle 

of   sliding    (see   Fig.    1)    is  45°.      The  weight  of  this  sliding 

1  The  facts  in  the  historical  outline  are  taken  from  "Neue  Theorie  des 
Erddruckes,"  E.  WINKLER,  Wien,  1872. 

2  Traite  de  la  defense  des  places. 

3  Traite  d'architecture  practique. 

1 


FIG.  1.— Method  of  Bullet. 


RETAINING  WALLS 


wedg?  ABC  i?  resolved  into  components  parallel  and  normal 
respectively  to  the  plane  of  slip.  The  former  component  was 
the  only  one  considered,  and  by  taking  moments  about  A, 
proper  wall  dimensions  are  found  to  resist  this  thrust.  COUPLET 
in  1727  makes  the  plane  of  cleavage  pass  through  the  outer  edge 
of  the  wall  (see  Fig.  2)  at  D.  The  prism  ACFE  is  resisted  by 
AED,  the  remaining  portion  of  the  wall  EBID  supporting  the 
wedge  EFB.  As  before,  the  weight  of  this  latter  wedge  EFB  is 
resolved  into  parallel  and  normal  components  and  the  former 
is  applied  directly  to  the  portion  of  the  wall  concerned.  To  get 

the  angle  that  the  plane  of 
cleavage  makes  with  the 
vertical,  he  followed  the 
method  of  MAYNiEL,1  tak- 
ing this  angle  equal  to  that 
of  the  slope  of  a  uniformly 
built  pile  of  shot,  the 
tangent  of  which  angle  is 

Vs. 

SALLONMEYER     and 

RONDELET      (1767)      follow 

the    method    of    Couplet, 


B 


FIG.  2.— Method  of  Couplet. 


save  that  the  plane  of  cleavage  starts  from  the  back  of  the 
wall.  BELiDOR,2  an  architect  formulated  a  method  in  which 
the  action  of  friction  is  considered.  Proceeding  as  in  the  above 
methods,  he  arbitrarily  assumes  that  one-half  of  the  wedge  weight 
is  consumed  in  overcoming  friction,  the  balance,  properly  re- 
solved into  parallel  and  normal  components,  acting  upon  the  wall. 

COULOMB  in  1774,  presented  the  first  rational  theory  making 
proper  allowance  for  friction  and  then  determining  the  wedge 
of  maximum  thrust.  Following  him,  NAVIER  and  finally 
PONCELET  developed  the  theory  into  its  present  form,  the  ele- 
gant graphical  method  of  determining  the  amount  of  thrust  be- 
ing due  to  the  latter. 

It  was  to  be  expected  that  the  brilliant  school  of  the  English 
and  French  mathematical  physicists  of  the  middle  of  the  last 
century  would  attempt  to  analyze  the  action  of  earth  pressure. 
Levy,  Boussinesq  and  Resal  of  France  and  Rankine  of  England, 

1  Traite  de  la  pousee  des  terres.     Memoire  publiee  dans    1'histoire  de 
1'academie  des  sciences,  1728. 

2  La  Science  des  Ingenieurs  L.  I.,  1729. 


THEORY  OF  EARTH  PRESSURE         3 

applied  the  methods  of  the  theory  of  elasticity  of  solids  to  granu- 
lar masses  with  varying  degrees  of  success.  Rankine's  results 
are  best  known.  Utilizing  the  so-called  ellipse  of  stress  (the 
stress  quadric  of  elastic  theory)  he  developed  his  theory  of  con- 
jugate pressures.  His  results  are  probably  the  most  universally 
applied  of  all  the  varied  methods. 

Later  analysts  of  earth  pressure  have  attempted  to  include 
in  the  theory  the  elements  of  friction  between  the  earth  and  the 
back  of  the  wall  and  that  of  cohesion  in  the  mass.  Such  attempts 
leave  intricate  expressions  of  decidedly  questionable  practical 
value. 

The  want  of  agreement  between  theory  and  experiment  has 
led  to  many  attempts  to  establish  empiric  relations  between  the 
width  of  the  wall  base  and  the  height  without  determining  the 
earth  thrust.  Sir  Benjamin  Baker,  the  illustrious  English 
engineer,  under  whose  supervision  the  London  tubes  and  outlying 
extensions  were  built,  advocated  a  value  of  this  ratio  of  about 
0.4,  one  which  Trautwine  warmly  seconds  in  his  handbook. 
Such  empiric  constants  were  of  value  when  walls  were  of  the 
rectangular  section,  or  verging  upon  the  revetment  type.  With 
the  modern  development  of  the  concrete  walls,  both  gravity  and 
reinforced  sections,  the  use  of  such  empiric  relations  is  decidedly 
questionable  and  good  engineering  practice  requires  that  a 
rational  method  of  ascertaining  the  wall  pressures  be  used  in 
determining  the  proper  dimensions  of  a  retaining  wall. 

Exact  Analysis  of  the  Action  of  Earth  Masses. — The  correct 
interpretation  of  the  character,  distribution  and  amount  of  pres- 
sures throughout  an  earth  mass  typical  of  ordinary  engineer- 
ing construction,  cannot  be  expressed  by  exact  mathematical 
analysis.  The  usual  earth  mass  retained  by  a  wall  contains  so 
many  uncertain  elements  (see  page  4)  that  can  neither  be 
anticipated  nor  determined  by  typical  tests,  that  it  becomes  very 
hard  to  assemble  sufficient  data  for  a  premise  upon  which  to 
found  any  satisfactory  conclusion.  To  analyze  an  earth  mass 
an  ideal  material  must  first  be  assumed.  The  divergence  in 
properties  between  that  of  the  actual  material  and  the  ideal 
material  determines,  in  a  more  or  less  exact  degree,  the 
approximation  of  the  results  found  theoretically. 

Under  such  uncertain  circumstances  and  with  a  consequent 
skepticism  of  mathematical  results,  the  natural  query  is — why 
attempt  a  refined  mathematical  analysis?  There  are  several 


4  RETAINING  WALLS 

praiseworthy  reasons.  The  general  action  of  earth  pressures 
may  be  indicated  and  reasonable  theories  may  be  advanced  as 
to  the  probable  character  of  pressures  to  be  anticipated.  A  good 
framework  may  be  built  upon  which  to  hang  modifications 
experimentally  determined.  The  several  mathematical  modes  of 
treatment  may  indicate  a  common  and  possibly  a  simple  expres- 
sion for  the  pressures,  of  easy  and  safe  application  to  most  of 
the  conditions  occurring  in  actual  practice.  Finally,  the  analysis 
of  the  ideal  earth  mass  may  show  the  maximum  pressures  that 
can  exist  in  the  usual  fills,  which  pressures  the  actual  ones  may 
approach  as  the  character  of  the  fill  approaches  that  of  the  ideal 
one  assumed.  Thus  the  probable  maximum  value  of  earth 
pressures  may  be  established;  an  important  function  and  an 
indication  of  the  probable  factor  of  safety  so  far  as  the  amount 
of  the  earth  thrust  is  concerned. 

The  Ideal  Earth  and  the  Fill  of  Actual  Practice. — The  mathe- 
matical discussions  of  the  action  of  earth  masses  premise  a  granu- 
lar, homogeneous  mass,  devoid  of  any  cohesion  (see  page  20)  and 
possessing  f  rictional  resistance  between  its  particles.  In  addition, 
the  surface  along  which  sliding  is  impending  is  assumed  to  be  a 
plane.  Such  a  fill  is  rarely  found  in  practice.  Fills,  ordinarily, 
are  made  either  from  balanced  cuts  for  street  or  railroad  grading, 
or  depend  upon  local  excavations.  In  the  usual  city  work, 
materials  for  fill  may  be  expected  from  other  local  improvements, 
public  or  private,  which  may  be  prosecuted  simultaneously,  or 
which  may  be  induced  to  be  prosecuted  because  of  the  expected 
place  of  disposal  for  spoil.  In  out  of  town  improvements  special 
steps,  such  as  the  employment  of  borrow  pits,  may  become 
necessary  to  provide  the  needed  material.  It  becomes  evident 
that  the  character  of  the  fill  may  vary  greatly,  containing  any  one 
or  several  types  of  earth,  and  including,  usually,  a  large  propor- 
tion of  excavated  rock. 

The  construction  of  the  embankment  itself  may  be  carried  out 
in  widely  different  manners.  It  may  be  built  up  from  a  tem- 
porary railroad  trestle,  the  materials  dumped  from  cars  and 
against  the  wall,  if  it  be  already  built.  Ordinary  teams,  or  motor 
trucks  may  dump  materials  upon  the  ground,  riding  over  the  fill, 
or  may  dump  over  the  slope  of  the  fill  already  formed.  Little 
homogeneity  can  be  expected  from  either  of  these  methods. 
Attempts  to  puddle  a  fill  to  give  it  eventual  compactness  and  in- 
creased density  make  it  difficult  to  team  over  the  puddled  portion 


THEORY  OF  EARTH  PRESSURE  5 

and  are  usually  abandoned  on  this  account.  While  specifications 
often  require  the  construction  of  an  embankment  in  thin  well- 
rammed  layers,  this  requirement  is  observed  more  often  in 
the  breach  than  in  the  observance.  It  is  a  costly  time-con- 
suming expedient  and  unless  required  by  special  types  of  design 
(see  page  21)  may  safely  be  ignored. 

Rarely  then,  in  either  the  type  of  the  earth,  or  in  the  mode 
of  utilizing  it  to  make  a  fill,  can  the  engineer  make  any  definite 
assumptions  as  predicated  for  the  ideal  earth,  nor  would  he  be 
justified,  from  the  standpoint  of  economy,  in  limiting  the  selection 
of  materials  for  fill  to  such  as  approach  the  character  of  the  ideal 
material,  especially  in  view  of  the  uncertainty  of  local  geologic 
conditions.  Obviously,  refinements  in  the  theory  of  earth  pres- 
sures and  attempts  to  predict  with  any  degree  of  exactness  the 
angles  of  repose  become  matters  of  more  or  less  academic  interest 
only. 

Bearing  in  mind  these  limitations  placed  upon  the  ideal 
material  assumed  in  the  following  analysis  and  that  the  mathe- 
matical work  is  developed  solely  as  a  means  toward  an  end,  as  was 
pointed  out  in  a  previous  page,  a  proper  appreciation  will  be  had 
of  the  relative  value  of  the  discussion  in  the  next  sections. 

The  Two  Theories. — The  theoretical  treatment  of  the  action  of 
earth  pressures  follows  along  two  fairly  distinct  lines.  The^ 
RANKINE  method  is  an  analytic  one,  starting  with  an  infinitesimal 
prism  of  earth  and  leading  to  expressions  for  the  thrust  of  the 
entire  earth  mass  upon  a  given  surface.  The  COULOMB  method, 
or  the  method  of  the  maximum  wedge  of  sliding  is  essentially  a 
graphical  one,  as  finally  shaped  by  PONCELET  and  treats  the 
mass  of  earth  in  its  entirety,  finding  by  the  principle  of  the 
sliding  wedge,  the  maximum  thrust  upon  a  given  surface.  It  will 
be  noticed  that  the  final  algebraic  expressions  for  the  thrust,  as 
determined  by  either  method,  are  similar  in  form,  and,  when 
certain  reasonable  modifications  (introduced  by  Prof.  William 
Cain)  are  placed  upon  the  Coulomb  method,  are  approximately 
alike  in  value  also. 

The  Rankine  Theory. — The  angle  of  internal  friction  (approxi- 
mately equal  to  the  angle  of  repose)  of  an  ideal  earth  as  defined 
above,  is  the  angle  0,  (see  Fig.  3)  which  the  resultant  force  R 
makes  with  the  normal  to  the  plane  when  sliding  along  this  plane 
is  just  about  to  start. 

In  a  mass  of  earth  unlimited  in  extent,  select  a  minute  triangular 


6 


RETAINING  WALLS 


prism,  whose  section  parallel  to  the  page,  is  a  right  angle  tri- 
angle, as  shown  in  Fig.  3.  In  addition,  let  the  prism  be  so 
selected  that  only  normal  stresses  exist  upon  its  arms.  These 
stresses  are  then  termed  principal  stresses,  and  the  planes  to 
which  these  stresses  are  normal,  are  termed  principal  planes.  The 
existence  and  location  of  such  planes  are  found  by  simple  methods 
given  in  the  text  books  on  applied  mechanics.  For  earth  masses, 
whose  upper  bounding  surfaces  are  planes,  Rankine  has  shown 
that  the  principal  planes  are  parallel  and  normal,  respectively  to 
the  upper  boundary  plane. 


FIG.  3. 


p  and  q  are,  respectively,  the  normal  stress  intensities  upon  the 
principal  planes  shown  in  Fig.  3. 

Since  there  is  a  limiting  value  of  the  angle  <f>,  which  limiting 
value  is  the  angle  of  repose,  or  better  termed,  the  angle  of  internal 
friction,  and  since  the  angle  i  of  the  triangular  prism  may  vary, 
it  is  possible  to  determine  a  maximum  value  of  0  for  some  value 
of  the  angle  i.  The  ratio  between  the  principal  stress  intensi- 
ties p  and  q  may  be  shown  to  be  independent  of  the  angle  i1 
and  can  be  denoted  by  some  constant.  With  the  value  of  the 
angle  <£  thus  defined,  it  is  possible  to  express  it  in  terms  of  the 
ratio  p/q,  since  the  angle  i  may  be  eliminated  after  its  value 
rendering  <£  a  maximum  is  found.  Knowing  the  maximum  value 
of  <£,  from  the  physical  properties  of  the  earth  in  question,  it  is 
thus  possible  to  express  the  stress  intensity  ratio  in  terms  of  the 

!See  Howe's  "Retaining  Walls,"  5th  Ed. 


THEORY  OF  EARTH  PRESSURE  7 

angle  <f>.     This  work  may  be  carried  out  by  utilizing  the  statics 
of  the  force  system  as  given  in  Figure  3. 
From  the  statics  of  Fig.  3 

tan  (i  -  <fi  =  Q/P 
and,  since     Q  =  qb,  and  P  =  pa, 

3-  +am    ?     Ql^*  =  g_jian  .J/Pi         smce  &/a  =  tan  *• 
P  Place  the  ratio  of  the  intensities  q/p  =  n. 
The  above  equation  then  becomes, 

tan  (i  —  0)  =  ?itan  i  (1) 

Denote  tan  i  and  tan  $  by  x  and  ?/  respectively,  and  expand  tan 
(i  —  4>)  by  the  formula 

,,         N         tani  —  tan  <£ 
tan  (i  —  4>)  =  7—        —  TT- 

1  +  tan  i  tan  0 

Equation  (1)  becomes 


By  the  principle  of  the  theory  of  maxima  and  minima,  this  ex- 
pression is  found  to  have  a  maximum  value  when  x  —  l/\/n. 
The  expression  for  y,  or  rather,  tan  <£,  for  this  value  of  x  is 

tan  0  =  ^^5  (3) 


To  reduce  this  to  the  form  as  finally  given  by  Rankine,  note  that 

tan  0 
sin  4> 


which  trigonometric  relation  reduces  (3)  to 

1  -  n 
sin*  =  Ff^ 

and  similarly 

q      1  -  sin  0 


This  gives  the  fixed  relation  between  the  principal  intensities 
of  stress  when  the  maximum  angle  of  friction  0  is  given,  and  the 
upper  surface  is  a  horizontal  plane.  The  value  of  the  principal 
intensity  p  upon  the  horizontal  plane,  is  easily  seen  to  be  the 
weight  of  the  earth  mass  above  this  plane.  If  the  depth  to  this 


8 


RETAINING  WALLS 


plane  is  h  and  the  unit  weight  of  the  material  is  w  then  p  =  wh 
and  (4)  becomes 

-   ,  1  —  sin  0 
q  =  wh  y—7- — -. (5) 

which  is  the  classic  relation  between  the  vertical  and  horizontal 
pressures  as  first  given  by  Rankine. 

This  is  the  fundamental  equation  of  the  Rankine  method  and 
the  following  theorems  are  deduced  directly  from  it:1 

(a)  The  direction  of  the  resultant  earth  pressure  against  a 
vertical  plane  is  parallel  to  the  free  upper  bounding  surface  and 
is  independent  of  the  interposed  wall. 

(6)  For  an  earth  mass  whose  upper  bounding  plane  makes  an 
angle  a  with  the  horizontal  (see  Fig.  4),  the  intensity  of  pressure 
parallel  to  CA  is 


I  —  wh  cos  a 


cos  a 


-v 


cos2  a  —  cos2  <f> 


cos  a  +  vcos2  a  —  cos2 


(6) 


This  expression  may  be  simplified  by  placing  cos  <£  /  cos  a  = 
sin  u  whence 

t  =  wh  cos  a  tan2  (u/2)  (7) 

Note  that,  in  this  expression,  tis  a  linear  function  of  the  depth 
of  earth  h,  so  that  the  value  of  the  entire  thrust  upon  a  plane 
A  B  of  depth  h  is 

T  =  th*/2.  (8) 

and  the  point  of  application  of  this  thrust  is  at  one  third  the  dis- 
tance h  above  B. 

(c)  The  final  resultant  thrust  upon  the  back  of  the  wall  BC  is 

compounded  of  the  above 
thrust  and  the  vertical  weight 
G  of  the  prism  ABC  (see  Fig.  4  j  . 
It  is  to  be  noted  that  no 
allowance  is  made  for  any 
frictional  resistance  that  may 


exist  between  the  back  of  the 
wall  and  the  earth  mass  im- 
mediately    adjacent     to    it. 
The  upper  surface  must  be 
free,    i.e.,   the    mathematical 
treatment   excludes    external 
loading  upon  the  upper  bounding  surface.     J.  Boussinesq  has 
1  HOWE,  "Retaining  Walls,"  5th  Ed.,  p.  11  et  seq. 


B 

FIG.  4. — The  Rankine  method  of  determin- 
ing the  thrust. 


THEORY  OF  EARTH  PRESSURE 


9 


attempted  to  extend  the  theory  of  Rankine  to  include  frictional 
action  between  the  earth  and  wall.1  The  complexity  of  his 
analysis  and  the  arbitrary  premises  although  of  the  utmost 
elegance,  preclude  its  acceptance  by  engineers.  In  fact,  it  is 
quite  doubtful  whether  the  Rankine  method  can  be  extended 
much  beyond  that  set  forth  above. 

The  average  earth  fill  has  an  angle  of  repose  approximately 
equal  to  30°.  As  pointed  out  on  page  4,  no  refinements  in  the 
selection  of  this  angle  are  justified  by  practical  conditions.  The 
expression  for  the  thrust  upon  a  vertical  plane  with  this  value  of  <f> 
becomes  with  t  =  wh/3 

T  =  w  —  (9) 

Taking  the  value  of  w  as  100  pounds  per  cubic  foot,  this  becomes 


T  =  16 


(10) 


For  a  wall  with  sloping  back  (the  usual  form  of  wall) ,  as  shown 
in  Fig.  5  the  thrust  is  found  by  combining  the  thrust  upon  the 
vertical  plane  AB  with  the 
weight  of  the  earth  over 
the  batter  of  the  back. 

The  upper  bounding  sur- 
face shown  in  Fig.  5  is 
that  typical  of  the  usual 
composite  fill  and  sur- 
charge equivalent  loads 
(see  later  pages  in  the 

chapter    for    a    full    discus-      FIG.  5.— Typical  loading  Rankine  method. 

sion  on  surcharges) .     Most 

retaining  walls  support  an  embankment  of  this  type.  For 
upper  surfaces  of  varying  types,  a  detailed  analysis  is  given  on 
pages  25  to  31. 

The  angle  of  friction  is  taken  at  30°,  with  the  consequent 
simplification  of  the  Rankine  formula.  The  ratio  of  the  height 
hf  to  h  is  denoted  by  c,  whence  the  total  depth  of  fill  acting  upon 
the  plane  AB,  Fig.  5,  is  h(l  +  c).  The  thrust  acting  upon  this 
plane  is  then 

P  =  wh2  (I  +  c)2/6. 

1  See  an  admirable  resume  of  his  work  in  this  direction  in  a  series  of  articles 
by  him  in  the  Annales  Scientifiques  deL'Ecole  Normale  Superiere,  1917  and 
reprinted  in  pamphlet  form  by  Gauthier-Villars,  Paris,  1917. 


10 


RETAINING  WALLS 


The  ratio  c  is  small,  generally  less  than  one-third,  whence  it 
is  permissible  to  substitute  1  +  2c  for  (1  +  c)2.  The  expression 
for  P  takes  the  form 


Note  here,  that  if  a  trapezoid  be  drawn  as  shown  in  Fig.  5  with 
ordinates  at  the  top  and  bottom  of  the  wall  the  earth  pressure 
intensities  at  these  points,  the  area  of  this  trapezoid  becomes 

2cl 
3 


wh2 


and  the  center  of  gravity  lies  at  a  point  Bh  above  the  base,  where 
B  has  the  value 

1  1  +  3c 


B  = 


31  +  2c 


(12) 


From  (11),  the  area  of  this  trapezoid  may  be  taken  equivalent 
to  the  thrust  upon  the  plane,  and  consequently,  equivalent  to 
the  horizontal  component  of  the  resultant  thrust  upon  the  back 
of  the  wall  AB.  The  thrust  is  located  at  the  center  of  gravity  of 
this  trapezoid  as  found  above. 

The  weight  of  the  earth  mass  superimposed  upon  the  back  of 
the  wall  is 


^          /L/I.  *      ,    ,   /^  tan  6  \  ,  2,      , 

G  =  w (hh  tan  o  H ~ )   =  w/rtjan  o 


1  +  2c 


(13) 


This  is  the  vertical  component  of  the  resultant  thrust  upon  the 
back  of  the  wall  and  the  value  of  the  thrust  T  is 

1         I       O- 

(14) 
(15) 


where  J  is  equal  to  -  V(l  +  9  tan26) 
o 


TABLE  1 


b° 

J 

0° 

b° 

/ 

6° 

0 

0.33 

0 

14 

0.42 

23 

2 

0.34 

4 

16 

0.44 

25 

4 

0.34 

8 

18 

0.47 

26 

6 

0.35 

11 

20 

0.49 

27 

8 

0.36 

15 

22 

0.52 

28 

10 

0.38 

17 

24 

0.56 

29 

12 

0.40 

21 

THEORY  OF  EARTH  PRESSURE 


11 


To  aid  in  the  computation  of  the  thrust  when  the  height  of 
wall  and  the  amount  of  surcharge  is  given,  as  well  as  the  slope  of 
the  back  of  the  wall,  Table  1  has  been  prepared  covering  a  number 
of  values  of  /  for  the  varying  values  of  the  angle  b. 

The  angle  which  the  thrust  T  makes  with  the  normal  to  the 
back  of  the  wall  is  (see  Fig.  5) 

0=  tan-1  (Q/P)  -  b  =  tan-1  (3  tan  6)  -  b  (16) 

from     equations     (11)     and     (13)     above. 

For  a  basis  of  comparison  with  the  formulas  developed  later, 
a  table  of  values  of  6  for  the  several  values  of  the  angle  b  is  given 
in  Table  1. 

To  summarize  briefly  the  results  above,  it  may  be  said  that 
equation  (14)  is  the  Rankine  expression  for  the  thrust  of  an  earth 
with  an  angle  of  repose  of  30°  whose  upper  surface  is  a  horizontal 
plane.  The  former  remarks  upon  the  usual  nature  of  embank- 
ments as  found  in  actual  practice  justify  a  blanket  assumption  of 
30°  for  this  angle  of  repose  and  the  resulting  simplification  of  the 
thrust  expression  strengthens  the  reasons  for  the  selection  of  that 
particular  value  of  the  angle  of  repose.  For  a  wall  with  sloping 
back  retaining  a  fill  of  shape  shown  in  Fig.  5  equation  (14)  gives 
the  expression  for  the  thrust.  The  computation  of  this  thrust  is 
to  be  aided  by  the  use  of  Table  1. 

Coulomb  Method  of  Maximum  Wedge  of  Sliding. — The  same 
assumptions  as  to  the  properties  of  the  ideal  earth  mass  are  made 
as  were  made  in  the  preceding  theory.  Referring  to  Fig.  6  any 


BT 


FIG.  6. — Method  of  maximum  wedge  of  sliding. 

prism  of  earth  AFC,  where  AC  makes  an  angle  a  with  the  hori- 
zontal, which  is  greater  than  the  angle  of  repose  <f>,  will  tend  to 
slough  away  from  the  remaining  Dearth  bank  and  will  therefore 
require  a  retaining  wall  with  back  AF  to  hold  it.  In  this  prism  of 


12  RETAINING  WALLS 

earth  the  forces  acting  upon  it  are  its  weight  G,  the  reaction  of 
the  thrust  T  upon  the  wall  and  the  reaction  of  its  pressure  Q 
upon  the  remaining  bank.  As  different  wedges  of  possible  sliding 
are  selected,  some  one  wedge  will  produce  the  maximum  thrust 
upon  the  wall  AF,  which  is  the  actual  thrust  sought. 

From  the  equilibrium  of  the  figure,  the  forces  T,  G  and  Q, 
are  concurrent,  i.e.,  must  meet  in  a  common  point.  From  the 
law  of  concurrent  forces 

T/sin  t  =  6r/sin  g  =  Q/sin  q.  t,  g  and  q  are  the  angles  as 
shown  in  the  figure. 

G  is  the  weight  of  the  irregular  prism  AFEC  and  is  resolved  by  the 
methods  of  equivalent  figures  (any  elementary  text  in  plane 
geometry)  into  the  triangular  prism  ABC.  If  a  slice  of  earth 
of  unit  thickness  is  taken  and  its  unit  weight  denoted  by  w,  the 
value  of  G  is 


A  T  is  normal  to  BC 
From  the  sine  relation  above  shown 


To  obtain  the  maximum  value  of  this  expression,  it  is  neces- 
sary to  separate  its  factors  into  those  which  remain  constant  as 
various  planes  of  sliding  are  selected,  and  those  which  vary  with 
the  different  planes  of  sliding.  This  is  effected  as  follows  : 

Draw,  in  Fig.  6,  what  may  be  termed  a  base  line  AZ  making 
an  angle  <£  +  <£'  with  the  normal  to  the  back  of  the  wall.  (The 
explanation  of  the  angle  <£'  will  be  given  later.)  Parallel  to 
this  line  draw  BO  and  CI.  In  ACI,  from  the  law  of  sines 

CI/AI  =  sin  t/  sin  g. 

(Note  in  the  figure  that  the  angles  g,  t  and  q  and  their  supple- 
ments are  denoted  by  the  same  letters. 
In  similar  triangles  CID  and  BOD 

CI/ID  =  BO/OD    and     BC/BD  =  OI/OD. 
Inserting  these  values  in  (18) 

_w        BD       Cl  _  w(ATXBDXBO\  ID  X  01  ,    } 
-~2A1ODU1AI"2\~        OD*  AI 

In  this  expression  all  factors  are  invariant  for  the  figure  except 

— 
AI 


the  factor  -  —  —  and  to  obtain  the  maximum  value  of  the 


THEORY  OF  EARTH  PRESSURE  13 

thrust  T,  it  is  sufficient  to  find  the  maximum  value  of  this  variable 
factor.  Upon  placing  A  I  =  x,  AD  —  a  and  AO  =  b,  introducing 
these  values  in  this  factor  and  then  proceeding  to  find  the  maxi- 
mum value  by  the  differential  calculus,  this  maximum  value  is 
found  to  occur  when 


x  =       W  (20) 

In  other  words  the  maximum  thrust  exists  upon  the  back  of  the 
wall  when  A  I  is  a  mean  proportional  between  AO  and  AD.  Fig. 
7  shows  a  simple  method  of  finding  a  mean 
proportional  by  geometric  construction. 
The  value  of  T  as  given  in  (19),  with  this 

,  ,  ,  ID  X  01          ,     ,  , 

new  value  of  the  term  --  —  =  —  may  further 

Al 

be  simplified  by  noting  that  triangles  DTA 
and  CHD  are  similar,  whence  AT/CH  — 
AD/CD;  BO/OD  =  CI/ID;  BD/OD  = 
CD  /ID.  Substituting  these  values  in  that 

1  .FiG.    7.  —  Geometric 

expression    for    the   thrust,    there   is   the    construction   for    mean 

Simple  form  proportional. 

T  =  ^  (CH  X  CI)  (21) 

L 

If,  with  I  as  a  center,  an  arc  CCf  is  described,  the  area  of  tri- 
angle CC'I,  multiplied  by  the  unit  weight  of  the  earth  is  equiva- 
lent to  the  maximum  thrust  T. 

The  direction  of  the  thrust  is  assumed,  in  the  original  method, 
to  be  normal  to  the  back  of  the  wall,  but  Prof.  Cain  has  modified 
this  so  that  the  direction  of  the  thrust  makes  an  angle  </>'  with  the 
normal  to  the  back  of  the  wall.  The  angle  <£'  is  the  angle  of 
friction  between  the  earth  back  of  the  wall  and  the  wall  masonry. 
(See  page  19  for  a  discussion  of  this  frictional  action  between 
earth  and  wall.) 

The  above  method  as  outlined  is  essentially  a  graphical  one  and 
in  order  to  make  a  comparison  between  the  results  of  this  method 
and  the  results  of  the  Rankine  method,  it  will  be  necessary  to 
obtain  an  algebraic  expression  for  the  thrust.  To  avoid  needless 
complications,  the  profile  of  the  earth  surface  will  be  assumed  to 
have  the  shape  shown  in  Fig.  8.  Without  entering  into  the 
tedious  but  quite  simple  steps  in  reducing  the  geometric  substi- 


DEPARTMENT  OF 


14 


RETAINING  WALLS 


tutions  above  to  algebraic  ones,  the  thrust  is  finally  found  to 
have  the  form 

T  =  |  VL  [l  +  c  -  pV(l  +c)2-c2/]2  (22) 

where 

cos  (#'  +  b)  , : — 


T 
- 


COS' 


sn 


vd 


/  =   tt/W 


u  -\-  v  tan  6 


6) 


sn 


cos  (</>'  +  6) 


d  =  tan  6  +  cot  i 
cos  (<£'  +  0  +  6) 


cos  (0'  +  6) 


h=ch 


C  D 

FIG.  8. — Typical  loading  Coulomb-Cain  Method. 

TABLE  2 


6° 

K 

b° 

K 

V  =  o° 

</.'  =  15° 

<t>'  =  30° 

0'  =  0° 

«'   =  15° 

0'  =  30° 

0 

0.33 

0.30 

0.29 

15 

0.45 

0.42 

0.43 

3 

0.36 

0.32 

0.32 

18 

0.4S 

0.45 

0.47 

6 

0.38 

0.34 

0.34 

21 

0.51 

0.48 

0.50 

9 

0.40 

0.37 

0.37 

24 

0.54 

0.52 

0.57 

12 

0.43 

0.40 

0.39 

When  the  back  of  the  wall  is  vertical,  i.e.,  6=0,  and  the  uppe, 
surface  is  horizontal  and  at  the  level  of  the  top  of  the  wall,  i.e. 
c  =  i  =  0,  the  expression  for  the  thrust  reduces  to 


«      1  -  sin  » 
2       1+  sin  4> 


(23) 


which  agrees  with  the  expression  obtained  on  page  7  using  the 
Rankine  method,  and  there  is  the  important  note  that  the  Thrust 


THEORY  OF  EARTH  PRESSURE  15 

upon  a  Wall  with  Vertical  Back  Due  to  a  Fill  Whose  Upper  Sur- 
face is  Horizontal  and  Level  with  the  top  of  the  Wall  is  found  to 
Have  the  Same  Expression  in  Both  Rankine  and  Coulomb  Methods. 
In  the  equation  for  the  thrust  (22),  the  term  c2/may  be  neglected 
and  as  before  the  term  (1  +  c)2  may  be  replaced  by  1  +  2c, 
whence  the  expression  takes  the  form 

±^  (24) 


K  =  L  (I  -  p)2.  Kis  finally  reduced  by  substituting  the  above 
values  of  m  and  p  in  it  and,  without  introducing  the  trigonometric 
steps,  is  given  by 


cos  (0'  +  b)  sin  <fr  sin 


L          /si 

[        "Vc 


cos2  (<£'  +  <£  +  b}  "cos  b  cos  (<*>'  +  6) 

To  compare  the  values  of  this  constant  K  with  the  constant 
of  parallel  meaning  /  found  on  page  10,  Table  2  has  been  pre- 
pared covering  a  range  of  values  of  b  and  $'.  As  before  the  value 
of  the  angle  of  repose  <f>  has  been  taken  as  30°. 

Note  that  if  in  Fig.  8,  the  trapezoid  ABCD  be  drawn  with  base 
Kwh  (  1  +  c)  and  ordinate  at  A  Kgch,  its  area  is 


which  is  equivalent  to  the  value  of  the  thrust  as  found  in  equation 
(24).  A  comparison  of  these  two  expressions  for  the  thrust, 
found  by  the  Rankine  and  by  the  Coulomb  method  and  a  study 
of  the  tabular  values  of  J  (Table  1)  and  K  (Table  2)  shows  the 
following  points: 

The  form  of  the  expression  giving  the  thrust  is  the  same  by  either 
method. 

For  values  of  the  angle  b  less  than  5°,  K  with  4>f  equal  zero  is 
the  same,  approximately,  as  J. 

For  values  of  the  angle  b  greater  than  5°,  K  with  4>r  equal  to  30° 
is  the  same,  approximately,  as  J. 

For  the  values  of  <£'  as  noted  in  the  preceding  the  directions  of 
the  thrusts  are  approximately  alike  using  either  theory. 
From  the  above  comparative  study  (also  see  examples  at  the  end 
of  this  chapter  giving  numerical  comparisons  of  thrust  computa- 
tion by  either  method)  it  is  seen  that,  with  the  limitations  as  shown 
above  (  see  pages  1  9  and  20  for  a  discussion  of  the  proper  values  of  the 


16 


RETAINING  WALLS 


angle  of  friction  to  be  assumed  between  the  back  of  the  wall  and 
the  earth)  either  of  equations  ( 14)  or  (24)  may  be  used  to  obtain  the 
value  of  the  thrust.  As  a  matter  of  fact  the  expression  as  deduced 
from  the  Rankine  equation  (14)  will  be  used  to  obtain  the  thrust, 
and  the  Coulomb  form  of  the  thrust  given  in  (24)  will  only  be 
used  where  its  form  lends  itself  more  readily  to  the  analysis 
of  the  special  problem  at  hand. 

To   recapitulate:  The   thrust   upon    any   wall    with    sloping 
back,  and  earth  profile  as  shown  in  Fig.  5,  is  to  be  found  from 


T  =  Jivh2 


2c 


where  J  is  the  earth  pressure  constant  to  be  taken  from  the 
values  of  J  found  in  Table  1,  c  is  the  surcharge  ratio,  and  w  is 
the  unit  weight  of  the  earth.  The  point  of  application  of  the 
thrust  is  located  at  a  distance  Bh  above  the  base  of  the  wall, 
where  the  values  of  the  ratio  B,  is  to  be  found  from  Table  3. 


TABLE  3 


c 

B 

c 

B 

c 

B 

0.0 

0.33 

0.5 

0.42 

1.0 

0.44 

0.1 

0.36 

0.6 

0.42 

1.5 

0.46 

0.2 

0.38 

0.7 

0.43 

2.0 

0.47 

0,3 

0.40 

0.8 

0.44 

Infinite 

0.50 

0.4 

0.41 

0.9 

0.44 

Admittedly,  neither  theory  meets  rigorously  the  application  of 
actual  conditions,  nor  are  they  confirmed,  experimentally  (see 
page  18  for  some  experimental  data  on  earth  pressures)  to 
any  great  degree  of  exactness.  It  follows,  then,  since  refinements 
are  not  only  unnecessary  but  superfluous  in  earth  pressure 
theories,  that  such  assumptions  and  approximations  as  have  been 
noted  and  applied  above,  should  suffice  for  all  retaining  wall 
design. 

It  is  essential  that  simplicity  of  thrust  calculation  be  kept  in 
mind,  as  it  is  by  far  more  important  that  a  standard  method 
of  such  thrust  determination  be  had,  than  that  the  refinements  of 
such  analysis  be  noted.  The  emphasis  upon  retaining  wall 
design  must  be  placed  upon  the  actual  design  of  the  wall  itself 
and  not  merely  upon  the  derivation  of  the  thrust. 


THEORY  OF  EARTH  PRESSURE  17 

As  a  matter  of  interest,  several  of  the  other  methods  of  thrust 
determination  are  given  in  the  following  section. 

Various  Methods  of  Thrust  Calculation. — Most  of  the  empir- 
ical expressions  for  the  thrust  have  the  form 

T  =  ch2  (26) 

with  various  assumptions  as  to  the  value  of  c.  On  page  9 
above,  the  value  of  c,  from  Rankine  and  from  Coulomb,  when 
the  angle  of  repose  0  is  taken  as  30°,  was  found  to  be  16. 

In  an  interesting  series  of  discussions  of  earth  pressures1 
this  value  of  c,  namely  16,  met  with  considerable  approval. 

The  analogy  between  lateral  and  hydrostatic  pressures  has 
been  utilized  in  some  formulas  by  assuming  the  earth  to  be  a 
fluid  with  unit  weight  varying  from  25  to  62  pounds  per  cubic 
foot,  the  latter  amount  supposedly  used  to  insure  a  satisfactory 
factor  of  safety.  These  assumed  weights  would  give  to  c  in  the 
above  empiric  equations  a  value  varying  from  12.5  to  31. 

C.  K.  Mohler,  in  the  Journal  of  the  Western  Society  of 
Engineers,  Vol.  15,  gives  a  modified  form  of  hydrostatic  pressure 
in  the  compromise  formula 

T  =  wh\l  -  sin  </>)/2  (27) 

where  w  is  the  unit  weight  of  the  material  and  $  is  the  so-called 
"  angle  of  flow."  He  states  that  the  lateral  earth  pressures  due 
to  earth  surcharges  is  probably  insignificant  and  illustrates  this 
by  an  ingenious  arrangement  of  cylinders.  Considerable  skep- 
ticism, however,  is  shown  in  regard  to  this  latter  statement  in  the 
discussions  on  his  paper,  and  doubtlessly,  the  author  of  the 
paper  has  not  credited  a  correct  effect  to  such  surcharges. 

In  Vol.  19  of  the  same  Journal,  a  modified  form  of  the  Rankine 
formula  is  given  and  is  urged  as  a  true  expression  for  both  lateral 
and  vertical  pressures. 

To  summarize  the  various  comments  upon  the  methods  of 
deriving  an  expression  for  the  earth  thrust,  it  may  be  stated  that 
although  objections  are  raised  to  practically  every  suggested  mode 
of  treating  such  pressures,  it  is  generally  conceded  that  retaining 
wall  failures  are  not  due  to  weaknesses  in  the  theory  of  pressures, 
but  are  primarily  due  to  faulty  design  and  construction.  This  is 
a  vital  conclusion  and  is  a  further  justification  for  the  adoption 
of  the  simple,  and  mathematically  sound,  expressions  given  in  the 

1  Western  Society  of  Engineers,  Vol.  16,  1911. 
2 


18  RETAINING  WALLS 

preceding  pages.  Examples  at  the  end  of  the  chapter  will 
illustrate  the  application  of  the  various  formulas  and  will  show 
the  simplicity  of  application  as  well  as  the  approximate  cor- 
rectness of  these  concise  expressions. 

It  may  be  stated  that  rule  of  thumb  methods,  both  for  the 
computation  of  the  earth  thrust  and  for  the  relations  between  the 
wall  dimensions  are  undesirable,  are  of  questionable  profes- 
sional practice  and,  in  the  case  of  reinforced  concrete  walls,  are 
not  only  inapplicable,  but  even  dangerous. 

Experimental  Data. — The  various  attempts  to  determine  earth 
pressure  values  experimentally,  have  been  quite  disappointing, 
so  far  as  definite  results  are  concerned;  but  they  have  led  to 
several  important  conclusions.  The  results  of  two  such  series 
of  experiments  are  given  here,  and  are  of  value,  not  only  for 
the  conclusions  reached  in  the  papers  themselves,  but  also 
because  of  the  summary  of  previous  experiments  given  therein. 

In  a  paper  by  E.  P.  Goodrich,  " Lateral  Earth  Pressures  and 
Related  Phenomena,"  Trans.  A.S.C.E.,  Vol.  liii,  p.  272,  the 
following  may  be  quoted  as  of  some  bearing: 

Sir  Benjamin  Baker  has  pointed  out  that  the  coarser  the 
materials  the  less  the  lateral  pressure. 

A.  A.  Steel.1  For  dry  and  moist  earth  the  lateral  pressure 
is  from  J£  to  J^  the  vertical  and,  in  saturated  materials  is 
practically  equal  to  it. 

Some  of  Mr.  Goodrich's  important  conclusions  are  as  follows : 

(a)  The  point  of  application  of  the  resultant  thrust  is  above  the 
%  point,  usually  about  0.4  of  the  height  of  the  wall. 

(6)  Rankine's  theory  of  conjugate  pressures  is  correct  when 
the  proper  angle  of  friction  is  found  (the  italics  are  mine),  and 
probable  adaptations  of  his  formulas  will  be  of  most  practical 
value. 

(c)  Angles  of  internal  friction  and  not  of  surface  slope  must  be 
used  in  all  formulas  which  involve  the  sliding  of  earth  over  earth. 
(Such  tables  are  to  be  found  in  the  author's  paper.) 

It  must  be  emphasized  that  the  experiments  mentioned  above 
were  performed  upon  a  more  or  less  homogeneous  material.  The 
actual  composition  of  fills  has  been  described  on  page  4. 

In  a  paper2  by  William  Cain,  the  conclusions,  after  analyzing 

Engineering  News,  Oct.  19,  1899. 

2  "Experiments  of  Retaining  Walls  and  Pressures  on  Tunnels,"  Trans. 
A.  S.  C.  E.  Vol.  Ixxii,  p.  403. 


THEORY  OF  EARTH  PRESSURE  19 

some  experiments  performed  by  the  author  and  analyzing  also  the 
extensive  experiments  carried  on  in  the  past,  are: 

"1.  When  wall  friction  and  cohesion  are  included,  the  sliding  wedge 
theory  is  a  reliable  one,  when  the  filling  is  a  loosely  aggregated  granular 
material,  for  any  height  of  wall. 

"2.  For  experimental  walls,  from  6  to  10  feet  high,  and  greater, 
backed  by  sand  or  any  granular  material  possessing  1  ttle  cohesion, 
the  influence  of  cohesion  can  be  neglected  in  the  analysis.  Hence 
further  experiments  should  be  made  only  on  walls  6  feet  and  preferably 
10  feet  high. 

"3.  The  many  experiments  that  have  been  made  on  retaining  walls 
less  than  one  foot  high  have  been  analyzed  by  their  authors  on  the 
assumption  that  cohesion  could  be  neglected.  This  hypothesis  is  so 
far  from  the  truth  that  the  deductions  are  very  misleading. 

"4.  As  it  is  difficult  to  ascertain  accurately  the  coefficient  of  cohesion, 
and  as  it  varies  with  the  amount  of  moisture  in  the  material,  small 
models  should  be  discarded  altogether,  in  the  future  experiments 
and  attention  should  be  confined  to  large  ones.  Such  walls  should  be 
made  as  light,  and  with  as  wide  a  base  as  possible.  A  triangular  frame 
of  wood  on  an  unyielding  foundation  seems  to  meet  the  conditions  for 
precise  measurements. 

"5.  The  sliding  wedge  theory,  omitting  cohesion,  but  including  wall 
friction,  is  a  good  practical  one  for  the  design  of  retaining  walls  backed 
by  fresh  earth,  when  a  proper  factor  of  safety  is  used." 

Clearly,  experimental  data  verifies  neither  of  the  above  theories 
with  any  degree  of  exactness,  yet  does  indicate  that  either  of  the 
two  theories  may  form  a  rational  basis  for  a  working  formula. 
Equation  (14)  may  again  be  brought  forward  as  the  practical 
formula  to  be  used  in  obtaining  the  thrust  upon  a  wall,  due  to 
the  usual  type  of  embankment  loading. 

The  above  work  has  frequently  discussed  the  items  of  wall 
friction  and  cohesion  and  these  two  factors  will  be  taken  up  in 
the  following  sections. 

Wall  Friction. — The  question,  whether  frictional  resistance 
between  the  back  of  a  retaining  wall  and  the  adjacent  earth  is, 
or  is  not,  a  permissible  factor  to  be  included  in  the  computation 
of  the  thrust  and  in  the  determination  of  its  direction,  plays  an 
important  role  in  various  theories  of  earth  pressure.  Since  the 
earth  backing  exerts  a  pressure  upon  the  wall,  then  by  the  ele- 
mentary theories  of  physics,  there  must  be  friction  between  the 
two  surfaces  in  contact.  The  angle  of  friction  cannot  be  assumed 
larger  than  the  angle  of  friction  of  the  earth  material,  since  if  it  is 


20  RETAINING  WALLS 

larger,  and  this  is  quite  possible,  the  effect  is  that  a  layer  of  earth 
will  adhere  to  the  wall  and  slipping  will  take  place  between  this 
layer  and  the  remainder  of  the  earth  bank.  If  allowance  is  made 
for  such  frictional  resistance,  it  is  customary  to  take  the  angle 
of  such  friction  (<£')  the  same  as  the  angle  of  repose.  This 
angle  has  been  taken  as  30°,  and  <£'  may  therefore  be  given  the 
same  value. 

The  question  of  lubrication  between  the  earth  and  wall  due 
to  the  presence  of  water,  must  be  taken  into  account  and  gener- 
ally the  more  vertical  the  wall  is,  the  greater  will  be  the  effect 
of  this  lubrication  upon  the  angle  of  wall  friction.  The  use  of 
equation  (14)  founded  upon  the  Rankine  method,  automatically 
provides  for  this  condition,  as  was  pointed  out  in  the  comparison 
between  the  Rankine  and  Coulomb  method  on  page  15. 

It  will  be  seen  later,  in  analyzing  the  various  types  of  walls, 
that  in  finding  the  proper  dimensions  of  a  gravity  wall  to  safely 
withstand  a  given  thrust,  quite  an  economy  in  the  necessary 
section  of  the  wall  is  effected  by  a  favorable  consideration  of  wall 
friction.  It  is  to  good  advantage,  then,  that  the  back  of  the  wall 
be  stepped  or  roughened  so  as  to  fully  develop  such  wall  friction. 

It  seems  better  engineering  practice  to  make  allowance  for 
such  a  force  than  to  ignore  it  and  assume  that  a  factor  of  safety 
of  unknown  value  is  thereby  added  to  the  wall.  Such  uncertain 
conditions  as  exist  in  wall  design  may  more  properly  be  allowed  for 
in  a  final  factor  of  safety  of  some  assumed  value,  than  to  merely 
add  blind  factors  by  ignoring  forces  which  must  surely  exist. 

The  question  of  wall  friction  plays  an  unimportant  role  in 
the  design  of  reinforced  walls  (whose  backs  are  usually  nearly 
vertical)  and  as  its  neglect  simplifies  the  calculation  of  the  wall,  it 
is  permissible  to  ignore  it — not  on  the  basis  that  it  does  not  exist, 
but  because  it  has  no  effect  upon  the  attendant  analysis. 

Cohesion. — Cohesion,  as  it  exists  in  an  earth  mass,  is  rather  a 
loosely  applied  term,  which  had  better  be  called  cohesional 
friction.  Prof.  William  Cain,  has  defined  its  action:1 

"The  term  'cohesive  resistance'  of  earth  may  properly  apply  either 
to  its  tensile  resistance  or  to  its  resistance  to  sliding  along  a  plane  in 
the  earth,  dependent  on  the  viewpoint.  However,  as  the  tensile  resist- 
ance of  the  earth  is  rarely  called  for,  the  term  'cohesive  resistance  of 
earth'  from  Coulomb's  time  to  the  present,  has  been  generally  restricted 
to  mean  the  resistance  to  sliding  as  affected  by  cohesion  *  *  *." 

1  Proc.  A.  S.  C.  E.  Vol.  xlii,  August,  1916,  p.  969. 


THEORY  OF  EARTH  PRESSURE  21 

To  properly  appreciate  the  effect  of  this  cohesional  friction, 
it  must  be  borne  in  mind  that  it  exists  to  some  extent,  varying 
from  a  slight  amount  to  a  very  large  amount,  in  all  earth  masses. 
It  is  the  one  element  that  probably  accounts  for  the  large  diver- 
gence between  theoretically  determined  and  experimentally 
determined  thrusts.  It  is  least  for  dry  granular  masses,  and 
reaches  a  maximum  value  in  the  plastic  clays. 

In  the  ordinary  fills  as  found  in  engineering  practice  (and 
over  90  per  cent,  of  walls  retain  embankments  of  fresh  fill)  its 
presence  is  a  highly  uncertain  one  and  in  view  of  the  mixed  char- 
acter of  such  a  fill  containing  boulders,  cinders  and  other  miscella- 
neous material,  its  existence  as  a  definite  resisting  force  to  sliding 
must  be  ignored.  General  practice  while  admitting  that  cohe- 
sion does  exist  in  earth  masses,  has  taken  the  very  wise  step,  to 
ignore  its  action.  While  this  may  increase  the  amount  of  thrust 
upon  a  wall,  it  is  very  possible  that,  due  to  vibrations,  or  other 
disturbances,  the  cohesive  action  in  the  earth  is  destroyed, 
temporarily  at  least  making  the  actual  thrust  approach  very 
closely,  in  value,  the  theoretical  thrust.  The  conclusions  of 
Prof.  Cain,  quoted  on  page  19  may  again  be  noted,  where  the 
method  of  the  sliding  wedge,  ignoring  cohesion,  is  recommended 
as  one  properly  determining  the  thrust. 

Under  certain  conditions,  where  a  direct  effort  is  made  to  obtain 
and  preserve  a  cohesive  effect  in  the  earth  mass,  it  is  within  rea- 
sonable practice  to  take  advantage  of  the  force.  When  a  wall 
retains  an  old  embankment,  where  only  a  thin  wedge  of  new  fill  is 
placed  between  the  old  fill  and  the  back  of  the  wall,  there  is  good 
justification  for  assuming  that  cohesion  will  be  a  permanent  force. 
Again,  by  carefully  placing  and  ramming  in  thin  layers  a  specially 
selected  fill,  cohesion  is  practically  assured  and  the  design  of  the 
wall,  may  safely  include  this  factor.  The  retaining  walls  of 
the  approach  to  the  Hell  Gate  Arch1  over  the  East  River,  New 
York  contain  a  fill  placed  with  extreme  care  and  the  determina- 
tion of  the  thrust  included  the  factor  of  cohesion,  permitting  the 
construction  of  a  fairly  thin  wall,  where,  under  ordinary  granular 
theory,  a  wall  of  prohibitive  section  would  have  been  required. 

The  effect  of  cohesion  may  be  interpreted  in  two  manners. 
It  has  been  noticed  that  the  bank  of  a  freshly  cut  trench  will  keep 
its  vertical  slope  for  quite  a  period,  and  then  as  it  sloughs  away 
will  gradually  approach  a  parabolic  shape,  with  the  upper  portion 

1  Engineering  News,  Vol.  73,  p.  886. 


22  RETAINING  WALLS 

more  or  less  vertical.  It  will  be  remembered  that  the  granular 
theories  above  discussed  have  assumed  that  the  surface  of  rupture 
is  a  plane.  To  allow  for  the  cohesive  action  as  described,  a  much 
steeper  angle  of  slope  for  the  material  may  be  assumed  than  its 
ordinary  angle  of  repose  would  warrant,  in  that  way  approaching 
the  parabolic  curve  or  it  may  be  assumed  that  for  a  certain  dis- 
tance below  the  surface  of  the  ground  there  is  no  lateral  pressure, 
the  surface  of  rupture  being  a  vertical  plane,  and  below  this 
critical  point  the  material  observes  the  ordinary  laws  of  the  granu- 
lar materials. 

The  first  method  is  an  empiric  one  and  seems  a  rather  perilous 
one  to  adopt,  in  view  of  the  uncertainty  of  cohesive  action.  The 
above  mentioned  retaining  walls  of  the  Hell  Gate  Arch  Approach 
were  designed  on  this  basis,  the  fill  taking  a  very  steep  slope.1 

TABLE  4 


Material 

c  in  Ibs.  per 
sq.  ft. 

Dry  sand                                                      .        

1.5 

Wet  sand 

8  3 

Very  wet  sand 

6  4 

Clayey  earth                                            .          

23.1 

Damp  fresh,  earth                                                                           .  . 

18.5 

Clay  of  little  consistency 

39  5 

A  theoretical  discussion  of  cohesion2  indicates  that  the  latter 
method  is  founded  on  more  logical  a  basis.  The  effect  of  cohe- 
sion is  to  lower  the  "head"  of  earth  pressure  so  that  a  soil  pos- 
sessing cohesion  exerts  no  lateral  pressure  until  a  certain  vertical 
pressure  has  been  reached,  corresponding  to  a  depth  #  in  the  earth. 
The  value  of  x  is  given  by  the  expression 

(28) 

c  is  the  coefficient  of  cohesion  for  the  material  and  may  be  taken 
from  Table  4.  w  is  the  unit  weight  of  the  material  and  $  is 
the  usual  angle  of  repose  of  the  material.  Below  this  depth  x,  the 
earth  pressures  follow  the  ordinary  laws  of  non-coherent  earths 
(see  Fig.  9).  An  application  of  the  above  formula  to  ordinary 

1  See  previously  quoted  article  in  Engineering  News. 

2  CAIN,  "Earth  Pressure,  Walls  and  Bins,"  p.  182  et  seq. 


THEORY  OF  EARTH  PRESSURE  23 

earth  with  some  cohesion  shows  that  this  lowering  of  the  head  is 
but  a  slight  one  and  for  all  practical  purposes  may  be  ignored. 
For  a  densely  compacted  material,  approaching  a  plastic  clay  this 
lowering  of  the  head  reaches  a  value  that  has  a  marked  effect 
upon  reducing  the  amount  of  the  thrust. 

In  an  interesting  paper  on  the  lateral  and  vertical  pressure  of 
clay1  a  set  of  formulas  for  the  stress  system  in  a  coherent  earth 
mass  was  given,  after  a  careful  experimental  study  of  the  neces- 
sary coefficients.  While  of  limited  application  (they  are  prima- 
rily for  the  clayey  materials)  they  are  worthy  of  quotation  and 
may  prove  of  service  in  interpreting  the  action  of  materials 


Ground 
Surface  • 


FIG.  9. — Coherent  earth. 

of  that  nature.  Before  presenting  these  equations  it  may  be 
well  to  note  the  character  of  some  of  the  stresses.  In  a  material 
more  or  less  plastic  there  is  a  tendency  for  the  surface  adjacent 
to  an  applied  loading  to  heave  and  raise.  This  may  be  shown  by 
a  mathematical  discussion  of  the  stress  distribution  in  a  material 
of  that  character2  and  is  clearly  demonstrated  by  experiment. 
Under  a  retaining  wall  the  pressure  is  generally  non-uniformly 
distributed,  having  a  maximum  value  at  the  toe  and  a  minimum 
value  at  the  heel.  From  the  foregoing  note  it  is  clear  that 
when  the  wall  bears  on  a  plastic  coherent  soil,  there  must  be  a 
certain  minimum  downward  pressure  at  the  heel  to  compensate 
for  the  upward  heaving  pressure  caused  by  the  soil  loading.  This 
is  given  below.  The  loading  which  a  soil  can  stand  without 
excessive  yielding  is  usually  termed  its  passive  stress,  as  distin- 
guished from  the  stress  which  it  exerts  (the  lateral  stress)  and 
which  is  termed  its  active  stress.  The  passive  stress  is  frequently 
called  the  ultimate  bearing  value  of  the  soil. 

1  BELL,  "Minutes  of  the  Proceedings  of  the  Institute  of  Civil  Engineers," 
Vol.  cxcix,  p.  233. 

2  See  HOWE,  5th  Ed.,  "Retaining  Walls." 


24 


RETAINING  WALLS 
TABLE  5 


Character  of  clay 

k  tons,  sq.  ft. 

a 

Very  soft  puddle  clay  

0  2 

0° 

Soft  puddle  clay  

0  3 

3° 

Moderately  firm  clay 

0  5 

5° 

Stiff  clay 

0  7 

7° 

Very  stiff  boulder  clay  

1  6 

16° 

The  retaining  wall  is  subjected  to  a  lateral  pressure  from  the 
coherent  material  of  intensity  pi,  which  is  given  by  the  equation 


-*  2k  tan        - 


Pi  =  wh  tan2 

(See  Fig.  9.)  a  and  k  are  the  constants  of  the  coherent  material, 
and  may  be  taken  from  Table  5.  From  the  above  expression 
it  is  to  be  noted  that  within  a  given  distance  x  below  the  surface, 
there  is  no  intensity  of  pressure.  This  value  of  x, 

x  =  ?*  Cot  (I  -  a\  (29) 

w  \4       27 

may  be  compared  to  the  similar  value  of  x  given  in  equation  (28) 
on  page  22. 

If  p2  is  the  minimum  permissible  intensity  of  downward  pres- 
sure on  the  foundation  at  the  heel  of  the  wall,  where  the  depth 
18  fl 

p2  =  wH  tan4  (7T/4  -  a/2)  -  2k  tan3  (ir/4  -  a/2)  -  2k  tan 

(7T/4  -  a/2)       (30) 

The  retaining  wall  rests  in  a  trench  and  its  footing  butts 
against  the  forward  part  of  the  trench  when  the  earth  pressure 
acts  upon  the  wall.  The  maximum  intensity  of  horizontal 
resistance  in  front  of  a  wall  at  any  depth  d  (note  that  this  is  a 
passive  stress)  is 

n  =  wd  tan2   (7T/4  +  a/2)  +  2k  tan  (ir/4  +  a/2)         (31) 

The  maximum  permissible  intensity  of  downward  pressure  on 
the  foundation  at  the  toe  of  the  wall,  where  the  depth  is  D  (note 
that  this  is  a  passive  stress,  usually  termed  the  safe  bearing  value 
of  the  soil)  is 

r2  =  wD  tan4  (*-/4  +  a/2)  +  2k  tan3  (n-/4  +  a/2)  +  2k  tan 

(7T/4  +  a/2)      (32) 


THEORY  OF  EARTH  PRESSURE  25 

While  the  above  series  of  equations  are  intended  primarily 
for  the  clays,  they  are  applicable  to  all  materials  upon  propel 
adjustment  of  the  values  of  the  coefficients.  Thus  for  non- 
coherent or  ordinary  granular  masses,  the  cohesion  coefficient  k 
is  zero  and  the  angle  4>  replaces  the  angle  a. 

In  a  discussion  upon  the  results  given  by  Bell,  Prof.  Cain 
has  noted,  that  if  A  is  the  value  of  a  unit  area,  then  the  relation 
between  the  k  given  here  and  the  c  of  his  material  is  k  =  cA . 

In  the  analysis  of  the  walls  in  the  following  chapters  and 
in  the  application  of  the  results  of  the  text  to  specific  problems 
the  action  of  cohesion  will  be  entirely  ignored,  the  formulas  given 
in  equations  (14)  and  (24)  being  used  to  obtain  the  thrust  upon 
the  wall. 

In  determining  the  strength  of  an  existing  wall  retaining  a 
well-settled  and  aged  embankment,  there  is  little  doubt  of  the 
existence  of  cohesion,  and  with  the  aid  of  the  preceding  equations 
a  proper  determination  of  the  load  carrying  capacity  of  the  wall 
may  be  obtained.  Whether  to  increase  the  load  upon  the  wall, 
by  addition  of  a  surcharge,  because  of  the  lowered  lateral  pressure, 
is  a  matter  of  judgment  and  in  view  of  the  uncertain  character 
of  cohesion  and  the  possibility  of  its  absence  for  some  unforeseen 
reason,  a  careful  engineer  may  sacrifice  apparent  economy  to  an 
easier  conscience. 

Surcharge. — While  a  surcharge  denotes  an  earth  mass  above 
the  level  of  the  top  of  the  wall,  it  is  customary  to  reduce  applied 
loadings  on  the  upper  surface  to  equivalent  surcharges.  In  the 
theory  of  the  distribution  of  stress  through  elastic  solids,  it 
has  been  proven1  that  such  distributions  are  independent  of  the 
manner  of  the  local  loading  except  for  points  fairly  close  to  such 
loads  and  it  is  permissible  to  substitute  the  resultant  load  for 
this  distribution,  or  conversely  a  distributed  loading  for  a  series 
of  concentrated  loads. 

It  seems  quite  justifiable  to  extend  this  law  to  granular  masses 
and,  in  fact,  it  is  generally  accepted  that  applied  loadings  may 
be  reduced  to  a  distributed  earth  surcharge  equivalent.  The 
reduction  of  dynamic  loadings  is,  possibly  more  involved  than 
that  of  the  reduction  of  still  loadings.  Nevertheless,  it  would 
seem  that  in  view  of  the  comparative  inelastic  properties  of  a 
granular  mass  and  of  the  large  amounts  of  voids  in  the  material, 

^See  for  example,  J.  BOUSSINESQ,  "On  the  Applications  of  the  Potential," 
etc! 


26  RETAINING  WALLS 

the  vibrations  are  completely  " dampened"  before  they  reach  the 
wall.  If  this  is  conceded,  no  distinction  need  be  made  between 
static  and  dynamic  loads.  In  any  event,  impact  coefficients 
of  as  great  value  as  are  applied  to  elastic  solids  should  not  be 
applied  to  the  earth  mass. 

While  there  may  be  some  question  as  to  whether  a  surcharge 
loading  produces  a  lateral  pressure  of  intensity  proportionate  to 
the  fill  proper,  below  the  level  of  the  top  of  the  wall  a  theoretical 
analysis  gives  no  foundation  for  such  doubt,  and  there  is  as 
tangible  a  basis  for  assuming  the  full  proportionate  effect  of 
the  surcharge  upon  the  wall  as  there  is  for  the  other  theoretical 
assumptions  of  earth  pressures. 

When  the  surcharge  is  uniformly  distributed  over  the  top 
of  the  embankment  and  extends  to  the  back  of  the  wall,  equations 
(14)  and  (24)  give  the  amount  and  Table  3  gives  the  location  of 
the  resultant  thrust.  When  the  surcharge  is  not  of  uniform 
distribution,  or  does  not  extend  to  the  back  of  the  wall,  the  con- 
ditions require  special  analysis.  The  following  treatment  of 
such  surcharges  is  given  primarily  for  the  same  reasons  as  in 
the  treatment  of  earth  pressures  in  general  and  is  to  be  used  in  the 
same  sense. 

When  an  external  loading  upon  an  embankment  has  been 
reduced  to  a  uniformly  distributed  loading  equivalent  to  the 
same  weight  of  earth,  a  new  profile  has  been  given  to  the  top  of 
the  embankment.  It  must  be  noted  here,  however,  that  when  a 
wedge  of  earth  is  about  to  slide  along  some  plane  in  the  fill  proper, 
this  plane  cannot  extend  at  the  same  slope  throughout  the  sur- 
charge, but  must  be  directed  vertically  upwards  after  reaching  the 
surface  of  the  ground  upon  which  the  surcharge  rests  (see  Fig. 
10).  The  method  of  the  maximum  wedge  of  sliding  is  most 
easily  applied  to  the  discussion  of  this  case  and  a  simple  graphical 
analysis  follows.1 

Let  the  equivalent  surcharge  extend  to  v.  Draw  a  line  parallel 
to  the  upper  surface  and  at  a  distance  2h'  above  it.  Draw  bn 
parallel  to  ov.  Connect  o  and  n.  The  intersection  s  of  this  line 
with  the  ground  surface  is  the  usual  base  point  to  construct 
the  equivalent  thrust  triangle.  Thus  through  s,  let  sa  be  parallel 
to  the  base  line  oz.  Locate  d  as  the  mean  proportional  between 
oA  and  oD,  and  locate  c  by  drawing  through  d  a  line  parallel 

1  Taken  from  MEHRTENS  "Vorlesungen  *****  Baukonstructionen"  as 
translated  by.G.  M.  PURVER,  Engineering  &  Contracting,  Nov.* 2,  191 0.1 


THEORY  OF  EARTH  PRESSURE 


27 


to  the  base  line.  Through  c  draw  uk  parallel  to  no.  With  d 
as  a  center  describe  an  arc  cm.  The  thrust  on  the  wall  due  to 
earth  and  the  surcharge  is  the  area  of  the  triangle  udm  multi- 
plied by  the  unit  weight  of  the  earth.  It  is  shown1  that  this 
triangle  is  equivalent  to  the  area  of  cdm  multiplied  by  the  ratio 
(h+2h')/h  =  l+2c  where  c  is  the  usual  surcharge  ratio. 
The  triangle  cdm  is  the  measure  of  the  thrust  upon  a  wall,  with 


FIG.  10. — Surcharge  not  extending  to  back  of  wall. 

no  surcharge,  whose  back  is  the  line  so,  making  the  angle  a  with 
the  vertical.  The  thrust  may  then  be  expressed  algebraically 
by 

T  =  &*l+M  K  (33) 

with  K  as  given  in  (25)  with  the  value  of  b  =  a.  When  the 
surcharge  extends  to  the  back  of  the  wall,  then  the  b  of  the  wall  is 
equal  to  a  and  the  form  for  the  thrust  in  this  case  is  the  same 
as  that  given  in  (24),  which  is  a  measure  of  the  approximation 
of  that  formula. 

To  determine  a  denote  the  distance  vb  by  r  and  let  this  be 
equal  to  yh.  Let  the  angle  voN  be  ft.  h  tan  ft  =  r  —  h  tan  b 
or  tan  ft  =  y  —  tan  b.  bm  =  2hf  tan  ft  =  2ch  tan  ft.  Nv  = 
h  tan  ft.  mN  =  r  —  bm  —  nv  =  h[y  —  (l+2c)  tan  ft],  tan  a  = 
mN  ii—  (l  +  2c}(y  —  tan  6)  _  fan  ^  _  2c 

~  1  +  2c  y' 

For  K  then  see 


(34) 


h(l+2c)  l+2c 

It  is  to  be  noted  that  a  may  be  negative. 
Table  13. 

The  application  of  the  wedge  of  maximum  thrust  to  the  case 

1  Ibid. 


28 


RETAINING  WALLS 


of  isolated  loads  on  the  surface,  is  quite  lengthy  and  involves 
considerable  geometric  construction.  It  is  discussed  fully  in 
the  lectures  mentioned  previously.  For  ordinary  practice  it 
seems  quite  sufficient  to  replace  it  by  its  equivalent  uniform 
spread  over  the  surface  and  then  to  apply  the  wedge  theory  to 
a  surface  of  broken  contour,  as  shown  in  Fig.  10. 

An  effective  and  simple  manner  of  treating  this  case  has  been 
devised  by  the  Design  Bureau,  Public  Service  Commission,  1st 
district  N.  Y.  and  is  as  follows: 

In  Fig.  11  there  is  a  concentration  of  L/a  as  shown,  a  surcharge 
of  h',  and  the  earth  back  of  the  wall.  For  some  plane  of  rupture 


FIG.   11. — Surcharge  concentrations. 

BN  all  three  exert  a  maximum  thrust  upon  the  wall.  A  few 
trials  are  ample  to  determine  this  plane  with  sufficient  accuracy.1 
Let  the  plane  of  maximum  thrust  make  an  angle  m  with  the 

horizontal.  The  thrust  TI  due  to  the  concentrated  load  is  — 
tan  (m  —  </>).  The  thrust  Tz  due  to  earth  and  surcharge  is 
o — —  cot  m  tan  (m  —  <£)  and  the  total  thrust  is  —tan 

-i  Qi 

ah2 
(m  —  0)  +  ^-  (1  +  c)2  tan  (m  —  <f>)  cot  m  the  maximum  value 

of  this  is  found  either  graphically  as  noted  above  or  by  equat- 
ing the  derivative  of  this  last  expression  to  zero,  whence,  upon 

<y/*2(l  +  c)2 

2 


placing  the  ratio  of  L/a  to 
sin  ' 


=  r 


-       cos  "" 


r  = 


sm2m 


-  cotm 


(35) 


1  See  CAIN,  "Earth  Pressure,  Walls  and  Bins,"  p.  43  for  an  excellent 
graphical  solution  of  this  case. 


THEORY  OF  EARTH  PRESSURE 


29 


Assuming  </>  =  30°  and  simplifying  the  expression 

sin  (2m  -  120°) 


(36) 


The  relation  between  m  and  r  is  shown  on  Curve  Plate  1.  When 
the  value  of  m  brings  the  wedge  of  thrust  inside  the  distribu- 
tion of  the  loading  L,  it  is  reasonably  certain,  unless  L  is  small, 
that  the  maximum  thrust  upon  the  wall  occurs  when  the  plane  of 


90 


80 


0) 

o> 

I 

CD 


70 


60 


O.I  0.2  0.3  0.4 

Ratio3  ~a 
Curve  Plate  No.  1. 

slip  just  encloses  the  spread  of  the  load  L.  Where  the  back  of 
wall  is  battered,  the  above  method  may  be  applied  to  the  ver- 
tical plane  through  the  heel  of  the  wall,  and  this  thrust  may  be 
combined  with  the  superimposed  weight  of  the  wall  over  the 
back. 

The  application  of  the  earth  and  surcharge  thrust,  if,  as  before, 
(1  +  c)2  is  replaced  by  1  +  2c,  (see  page  15)  is  at  the  center  of 
gravity  of  the  trapezoid  of  loading,  or  at  a  distance  Bh  above  the 
bottom  of  wall,  with  B  as  given  in  Table  3.  The  thrust  due  to 
the  isolated  load  may  be  assumed  to  be  distributed  uniformly 
along  the  back  of  the  wall,  from  the  base  of  such  load  to 


30  RETAINING  WALLS 

the  bottom  of  wall.  As  shown  in  Fig.  10  its  lever  arm  is 
then  C/2. 

A  simple  method  of  reducing  isolated  concentrated  loads  to  a 
uniformly  distributed  surcharge,  making  the  standard  thrust 
equations  ( 14)  and  (24)  applicable  is  as  follows.  The  concentrated 
load  is  assumed  to  be  transmitted  along  slope  lines  making  an 
angle  of  30°  with  the  vertical.  (See  the  following  pages  of  this 
chapter  for  the  experimental  justification  of  this  assumption.) 
At  the  point  I,  where  this  distribution  strikes  the  line  AB,  see 
Fig.  11,  determine  the  intensity  of  vertical  pressure.  With  this 
as  the  new  surcharge  equivalent,  employ  the  above  equations 
to  determine  the  thrust.  This  method  is,  of  course,  quite  ap- 
proximate, and  should  be  used  more  as  a  method  of  confirming 
the  results  obtained  in  the  more  exact  construction  above,  than 
as  a  primary  method  of  getting  the  thrust.  An  example  at  the 
end  of  this  chapter  will  illustrate  the  two  methods. 

The  preceding  discussion  of  surcharge  loadings  has  confined 
itself  to  the  lateral  effect  of  such  loadings  upon  a  retaining  wall. 
It  may  be  of  interest  to  determine  the  vertical  intensity  of  such 
loadings  at  distances  below  the  upper  bounding  surface.  The 
intensity  diminishes  as  the  distance  from  the  upper  surface  in- 
creases and  its  spread  may  be  said  to  be  confined,  roughly, 
within  the  surface  of  a  cone.  Several  expressions  are  given  for 
the  intensity  at  any  plane  below  the  upper  surface. 

In  Vol.  20,  Journal  of  the  Western  Society  of  Engineers,  Mr. 
Lacher  has  given  the  following  expression  for  the  vertical  live- 
load  intensity  at  any  depth  h  below  the  surface  (due  to  locomotive 
wheel  loads) 

11000 

where  x  is  the  inclination  of  the  spread  planes  in  fractions  of  a 
foot  per  foot  of  depth. 

The  distribution  of  pressure  through  soil  has  been  experimen- 
tally determined1  and  for  depths  of  over  3  feet  there  is  a  spread  of 
fairly  uniform  intensity  extending  within  slope  planes  making  an 
angle  of  30°  with  the  vertical. 

An  empiric  expression  given  by  Prof.  Melvin  L.  Enger  in  the 
Engineering  Record  Jan.  22,  1916,  p.  107,  for  the  intensity  of 

1  Proc.  Am.  Soc.  Testing  Materials,  Vol.  17,  part  2,  1917. 


THEORY  OF  EARTH  PRESSURE  31 

vertical  pressure  at  any  depth  as  experimentally  determined  is  as 
follows : 

A  =  pB 

where  A  is  the  intensity  of  pressure  at  a  depth  h  in  inches,  B 
is  the  surface  intensity  of  pressure  and  p  is  the  percentage  of 
the  surface  intensity  given  by  the  following 

p  =  91  dlM/h1-** 

The  authors  of  the  paper  doubt  whether  .the  above  expression  has 
general  application.  It  would  show,  roughly,  however,  that  such 
transmitted  pressure  varies  as  the  inverse  square  of  the  distance 
below  the  loaded  surface.  A.  E.  H.  Love  has  shown1  that  the 
transmitted  pressure  through  an  isotropic  solid,  at  a  distance 
h  below  the  loaded  surface  and  directly  below  the  loaded  point  is 

3TF   1 

"27  F* 

so  that  there  is  a  striking  agreement  in  the  variation  of  trans- 
mitted pressure  in  solid  and  granular  masses.  For  an  interesting 
treatise  on  the  distribution  of  pressure  through  solids  for  any 
character  of  surface  loading,  See  " Application  des  Potentials" 
by  J.  Boussinesq,  pp.  276  et  seq. 

Pressure  on  Cofferdams. — A  cofferdam  retaining  earth  is  in  a 
sense,  a  retaining  wall  subject  to  the  ordinary  theory  of  lateral 
pressures.  The  cofferdam  itself  is  an  assembly  of  sheeting,  wal- 
ing pieces,  or  rangers  and  braces,  the  design  of  which  follows  the 
ordinary  theory  of  the  design  of  timber  structures.  Mr.  F.  R. 
Sweeny2  has  presented  a  thorough  investigation  of  the  loadings 
upon  such  a  structure  together  with  a  study  of  the  economics  of 
its  design. 

His  design  has  been  predicated  upon  the  assumption  that  the 
ratio  of  the  unit  lateral  pressure  to  the  unit  vertical  pressure 
is  given  by  a  constant  c  (corresponding  to  the  earth  pressure 
coefficients  K  and  J  of  the  preceding  pa*ges) .  The  unit  weight  of 
the  material  outside  the  sheeting  is  denoted  by  w.  To  quote  the 
author  : 

"  The  values  of  w  and  c  are  not  easily  determined  being  largely  matters 
of  mature  judgment.  In  any  event,  it  is  important  to  look  into  the 

1  "A  Treatise  on  the  Mathematical  Theory  of  Elasticity,"  1st  Ed.,  p.  270. 

2  Engineering  News-Record,  April  10,  1919,  pp.  708  et  seq. 


32  RETAINING  WALLS 

matter  of  possible  saturation  of  the  soil  to  the  point  where  hydrostatic 
pressure  will  be  developed  and  superimposed  upon  the  earth  pressure." 

The  economic  proportions  and  the  best  dimensioning  of  the 
timbers  and  sheeting  (wood  and  steel)  are  given  in  the  article  and 
the  entire  design  is  exhaustively  treated. 

Pressures  of  Saturated  Soils. — With  the  presence  of  water  in 
a  soil,  an  additional  lateral  pressure  is  exerted  from  the  plane  of 
the  water  surface  to  the  bottom  of  the  wall.  An  interesting 
paper  by  A.  G.  Husted1  discusses  in  detail  this  important  ques- 
tion. The  following  quotations  from  the  paper  cover  the  salient 
features  of  the  treatment. 

"Formulas  giving  the  lateral  pressure  of  earth  against  vertical 
walls  may  be  found  in  many  text  books  and  hand  books.  These  for- 
mulas, however,  usually  refer  to  dry  earth  and  not  to  earth  which  is 
saturated  with  water.  The  writer  has  had  occasion  when  designing 
structures,  wholly  or  in  part  below  water  level  to  calculate  the  lateral 
pressure  of  saturated  earth,  and  being  unable  to  find  a  satisfactory 
method  for  computing  these  pressures  has  worked  out  the  method 
herein  set  forth." 

The  writer  of  the  paper  states  that  he  will  apply  the'  RANKINE 
relation  between  the  lateral  and  vertical  intensities  as  given  by 
equation  (14). 

"  As  has  been  noted  before,  the  formula  assumes  that  the  lateral  pres- 
sure at  any  point  bears  a  definite  relation  to  the  vertical  pressure,  this 
relation  depending  entirely  upon  the  angle  of  repose.  It  will  thus  be 
seen  that  the  second  part  of  the  equation  can  be  divided  into  two  parts, 
wh  representing  the  unit  vertical  pressure  and  (1  —  sin  $)/(  1+  sin  <f>) 
representing  the  relation  between  lateral  and  vertical  pressures. 

"Two  methods  of  applying  this  formula  to  cases  involving  saturated 
earths  have  been  and  are  still  in  quite  general  use.  One  of  these 
methods  consists  in  computing  the  total  lateral  pressure  in  the  usual 
way  using  for  w  the  weight  #f  dry  earth  and  for  <£  the  angle  of  repose 
of  dry  earth.  To  this  pressure,  then,  is  added  full  hydrostatic  pressure 
below  the  plane  of  saturation.  This  method  may  quite  often  give 
results  close  enough  to  actual  conditions  for  ordinary  purposes  of  design, 
but  it  appears  to  the  writer  to  be  at  variance  with  the  fundamental 
formula.  In  the  first  place,  no  allowance  is  made  for  the  fact  that  satu- 
rated earth  has  a  smaller  angle  of  repose  than  dry  earth,  and  in  the 

1  Engineering  News-Record,  Vol.  81,  p.  441  et  seq. 


THEORY  OF  EARTH  PRESSURE 


33 


second  place  it  is  assumed  that  earth  weighs  the  same  in  water  as  it 
does  out  of  water. 

"Another  method  of  calculating  lateral  earth  pressures  consists  in 
computing  the  total  lateral  pressure  in  the  ordinary  way  and  adding 
to  this,  partial  hydrostatic  pressure  below  the  plane  of  saturation.  The 
amount  of  the  partial  hydrostatic  pressure  is  determined  by  taking  the 
difference  between  full  hydrostatic  pressure  and  lateral  earth  pressure 
for  an  equivalent  depth.  This  method,  however,  can  easily  be  proved 
erroneous  by  applying  it  to  a  fill  of  completely  saturated  earth.  In 
this  case  the  partial  hydrostatic  pressure  to  be  added  will  be  the  difference 
between  full  hydrostatic  pressure  and  lateral  earth  pressure  for  the  total 
depth  of  earth.  It  can  thus  be  seen  that  the  total  lateral  pressure  at 
the  bottom  would  be  exactly  equal  to  full  hydrostatic  pressure.  This  is 
absurd. 

"In  order  to  correct  the  errors  in  the  above  mentioned  methods,  a 
method  has  been  worked  out  which  the  writer  believes  to  be  theoretic- 
ally correct.  In  this  method  the  following  assumptions  are  made : 

Lateral  earth  pressure  varies  directly  with  the  vertical  earth  pressure  for 
earth  with  any  given  angle  of  repose  and  is  equal  to  the  vertical  pressure 
multiplied  by  (1  —  sin  </»)/(!  +  sin  <£). 
Water  exerts  full  hydrostatic  pressure 
laterally  as  well  as  vertically  regard- 
less of  the  amount  of  the  space  oc- 
cupied by  earth. 

"It  is  a  well  known  fact  that  the 
angle  of  repose  of  earth  in  water  is 
less  than  the  angle  of  repose  of 
dry  earth.  Therefore  the  ratio  of 
lateral  pressure  to  vertical  pressure 
is  greater  below  the  plane  of  satu- 
ration than  above.  On  page  580 
of  Merriman's  "American  Civil 
Engineers'  Pocket  Book"  the  angle 
of  repose  of  dry  earth  is  given  as 
36°53'  while  that  of  soil  under  water  is  given  as  15°57/. 

"Above  the  plane  of  saturation  the  lateral  pressure  is  computed  in 
the  usual  manner.  Below  the  plane  of  saturation  the  lateral  pressure 
is  obtained  by  multiplying  the  total  vertical  pressure  less  the  buoyant 
effect  of  water  by  (1  —  sin  <£)/(!  +  sin  $)  and  adding  to  this  the  full 
hydrostatic  pressure.  For  example,  in  Fig.  12  the  unit  lateral  pressure 
pa  at  point  a  which  is  above  the  plane  of  saturation  is  Wih(l  —  sin  <J>)/ 
(1  +  sin  $).  MI  is  the  weight  of  the  dry  earth  per  cubic  foot,  h  is  the 
distance  of  the  point  a  below  the  surface  and  <£  is  the  angle  of  repose 
of  dry  earth.  Likewise  the  unit  lateral  pressure  pb  at  point  b  below  the 
plane  of  saturation  is  (wihi  +  ^2^2)  (1  —  sin  <£)/(!  +  sin  <j>)  +  62.5  hz. 

3 


FIG.  12. 


34  RETAINING  WALLS 

wi  as  above  is  the  weight  of  the  dry  earth  per  cubic  foot,  hi  is  the  distance 
from  the  ground  surface  to  the  plane  of  saturation,  wz  is  the  weight  per 
cubic  foot  of  earth  under  water,  h2  is  the  distance  of  the  point  b  below 
the  plane  of  saturation  and  </>2  is  the  angle  of  repose  of  earth  under  water. 
"It  will  be  noticed  that  in  this  method,  for  points  below  the  plane 
of  saturation,  hydrostatic  pressure  and  earth  pressure  are  separated; 
that  full  hydrostatic  pressure  is  allowed;  that  the  vertical  pressure  is 
obtained  by  adding  the  total  weight  of  earth  above  the  plane  of  satura- 
tion to  the  net  weight  (weight  under  water)  of  earth  below  the  plane 
of  saturation;  that  the  lateral  earth  pressure  is  obtained  by  multiplying 
the  vertical  pressure  by  (1  —  sin  <£2)/(l  +  sin  <£2) ;  that  the  total  lateral 
pressure  is  obtained  by  adding  the  hydrostatic  pressure  to  this  lateral 
earth  pressure. 

"It  can  be  readily  seen  that  if  a  smaller  angle  of  repose  is  assumed 
for  saturated  earth  than  for  dry  earth,  there  will  be  a  decided  increase 
in  the  unit  lateral  pressure  at  the  plane  of  satura- 
tion. In  other  words,  the  unit  lateral  pressure  an 
infinitesimal  distance  below  the  plane  of  saturation 
will  be  much  greater  than  that  at  an  infinitesimal 
distance  above  the  plane  of  saturation. 

"  At  first  thought  this  appears  absurd,  but  it  can 
be  seen  that  it  should  be  so.     It  can  perhaps  be 
^^  best  illustrated  by  an  exaggerated  example.     Take 

the  case  of  a  retaining  wall  supporting  a  bank  of 
earth  loaded  with  timbers  (Fig.  13),  the  lateral  pressure  of  the  timbers 
against  the  wall  is  zero,  but  at  an  infinitesimal  distance  below  the 
surface  of  the  earth  the  pressure  is  a  considerable  amount  due  to  the 
load  that  is  superimposed. 

"The  difference  is  plainly  due  to  a  difference  in  the  angle  of  repose." 

While  the  preceding  analysis  is  a  correct  mathematical  interpre- 
tation of  the  action  of  saturated,  homogeneous  material,  devoid 
of  cohesion,  and  may  be  used  with  the  same  degree  of  freedom  as 
any  of  the  carefully  worked  out  theories  of  earth  pressure,  it  is 
open  to  the  same  vital  objections  as  were  stated  on  the  pages 
preceding.  However,  as  long  as  a  proper  appreciation  is  had  of 
the  limitations  of  theory  in  general  and  if  the  lateral  pressures 
are  computed  as  suggested  on  page  16  and  as  given  by  the 
equations  there  shown  the  method  presented  by  Mr.  Husted  is  a 
practical  one  and  should  be  followed  provided  a  safe  lateral  thrust 
of  saturated  soils  is  sought. 

Sea  Walls. — A  sea  wall  is  essentially  a  retaining  wall  with  a 
fill  of  varied  character  behind  it,  composed,  usually  of  rip-rap, 


THEORY  OF  EARTH  PRESSURE  35 

earth,  cinders  and  the  like,  and  subject  to  a  hydrostatic  pressure 
varying  with  the  tide.  An  analysis  of  the  pressure  to  which  sea 
walls  are  subjected  is  given  in  an  article  byD.  C.  Serber,  Engineer- 
ing News,  August  23,  1906,  excerpts  of  which  are  quoted  below. 
Walls  with  vertical  backs  are  the  only  type  treated.  The  Rank- 
ine  method,  as  applied  in  the  previous  pages,  is  used  in  this 
treatment,  the  thrust  intensity  being  given  by  equation  (5). 
It  is  assumed  in  the  paper  that  the  fill  varies  by  strata,  a  hori- 
zontal plane  separating  the  fills  of  different  character.  If  the 
fill  back  of  the  wall  is  assumed  to  be  composed  of  two  such  mate- 
rials, of  weights  Wi  and  wz,  respectively  and  separated  from  each 
other  by  a  horizontal  plane,  hz  above  the  bottom  of  the  wall  and 
hi  below  the  top  of  the  wall  Mr.  Serber  notes  the  following  im- 
portant conclusion  (theoretically  deduced)  : 

"  The  total  pressure  on  the  lower  section  of  the  wall  (i.e.,  below  the 
plane  of  separation)  is  entirely  independent  of  the  angle  of  natural 
repose  of  the  material  above  the  plane  of  separation." 

If  the  angle  of  repose  of  the  upper  material,  of  weight  Wi  is  </>i 
and  that  of  the  lower  material,  of  weight  w2,  is  $2  and  if,  for  the 
sake  of  simplifying  the  resulting  expression  there  is  put 

m  =  hi/hz',  n  =  Wi/wz  and  ai  =(^0°  ~~  #0  fl2  =       (90°   —   <£2) 


the  total  pressure  P2  on  the  back  of  the  wall  is 


An  ingenious  graphical  method  of  obtaining  the  total  pressure 
of  two  or  more  layers  of  different  fill  is  presented  in  the  paper 
founded  upon  the  reduction  of  the  different  weights  in  terms  of 
one  of  the  weights. 

The  effect  of  surcharge  upon  a  sea  wall  is  discussed  as  follows  : 

"Merchandise,  cranes  and  other  loads  of  considerable  weight  are  apt 
to  be  stored  temporarily  or  permanently  on  the  sea  wall  and  the  backing 
immediately  behind  it.  The  Department  of  Docks  and  Ferries  of 
New  York  City  assumes  a  uniform  vertical  load  of  1000  pounds  per 
square  foot,  *  *  *.  When  the  bottom  is  very  soft  mud  of  consider- 
able depth  and  a  pile  foundation  is  to  be  resorted  to,  the  normal  dif- 
ficulties of  sustaining  a  retaining  wall  are  so  great  that  it  becomes 
highly  desirable  to  avoid  the  additional  thrust  due  to  the  surcharge. 
In  such  cases  a  platform  may  be  built  *  *  *  supported  on  an  in- 
dependent foundation  sufficient  to  carry  the  surcharge,  thus  relieving 
the  wall  of  the  thrust  *  *  *." 


36  RETAINING  WALLS 

The  inclusion  of  hydrostatic  pressure  upon  this  wall  may  be 
dealt  with  in  the  manner  outlined  in  the  preceding  section,  the 
formulas  of  Mr.  Berber  being  readily  adaptable  to  the  principles 
given  in  that  section. 

It  must  be  emphasized  that  a  sea  wall  is  a  structure  of  peculiar 
importance  in  the  design  of  which  the  paramount  question  is  not 
one  of  ascertaining  how  great  the  thrust  upon  its  back  is,  but 
how  can  its  foundation  carry  the  loads  brought  upon  it.  Accord- 
ingly due  appreciation  to  this  question  must  be  given  before 
attempting  refinements  in  the  calculation  of  the  thrusts  that  may 
be  induced  in  the  wall  by  the  fills  deposited  behind  it. 

A  number  of  problems  have  been  prepared  at  the  end  of  this 
and  the  succeeding  chapters  to  illustrate  the  application  of  the 
several  tables,  curves  and  equations  given  in  the  text  immediately 
preceding.  They  will  also  serve  to  demonstrate,  numerically, 
the  points  discussed  in  the  chapter,  bringing  home  more  forcibly 
the  truths  quoted  than  the  literal  equations. 

Problems 

1.  A  wall  with  a  back  sloped  to  a  batter  of  one  on  four  and  30  feet  high 
supports  a  level  fill  subject  to  a  surcharge  loading  of  600  pounds  per  square 
foot.  What  are  the  thrusts,  by  both  Rankine's  and  Coulomb's  methods 
(a)  when  there  is  no  surcharge;  (6)  when  the  surcharge  extends  to  the  wall 
a  (see  Fig.  5);  (c)  when  the  surcharge  extends  up  to  the  point  6,  directly 
over  the  heel  of  the  wall.  • 

The  angle  that  the  back  makes  with  the  vertical  is  tan-1  (^)  =14°.  For 
the  condition  of  no  surcharge,  from  (14)  and  Table  1  with  J  =  0.42  for  b  = 
14°. 

T  =  10°  *  3°2  X  0.42  =  18,900  pounds. 

4t 

From  Table  1,  0  =  23°  and  the  angle  that  the  thrust  makes  with  the  hori- 
zontal is  23°  +  14°  =  37°. 

From  (25)  and  Table  2  for  <£'  =  0°,  15°  and  30°,  K  =  0.44,  0.41  and  0.42 
respectively  and  the  values  of  the  thrusts  are  accordingly,  19,800,  18,500 
and  18,900  pounds. 

For  the  condition  of  the  surcharge  extending  to  the  back  of  the  wall, 
the  constants  remain  as  above  and  since  c  =  %o  =  0-2,  the  thrusts  are 
each  increased  by  (1  +  2c)  or  by  1.4.  The  thrust,  using  Rankine's  method 
is  then  1.4  X  18,900  =  26,500  pounds.  The  three  thrusts,  employing  the 
method  of  the  sliding  wedge  method  become,  respectively  27,800,  25,900 
and  26,500^pounds  as  the  angle  of  friction  between  wall  and  earth  is  taken 
as  0°  15°  or  30°. 

When  the  surcharge  extends  to  6  the  condition  under  which  the  method 


THEORY  OF  EARTH  PRESSURE 


37 


of  Rankine  is  used  must  receive  special  investigation,  since  equation  (14) 
no  longer  applies.     From  (11)  with  c  =0.2,  the  thrust  is 
100  X  302  X  1.4 


T  = 


=  21,000 
30  X  K 


7.5,  11,250  pounds 


2X3 

The  weight  of  the  triangle  G  is,  since  db 
and  the  resultant  thrust  upon  the  wall  is 

To  =  \/(21,000)2  +  (11,250)2  =  23,700  pounds. 
The  angle  which  this  final  thrust  makes  with  the  horizontal  is 
tan-1  (11,250/21,000)  =  28°. 

With  the  expression  given  in  (33),  the  method  of  the  sliding  wedge  may 
be  employed,  after  the  proper  value  of  a  has  been  found.     The  value  of  the 

04 
ratio  y  is  7.5/30  =  0.25.     From  (34)  tan  a  =  0.25  -  y^  0.25  =  0.18,  from 

which  a  =  10°  and  the  corresponding  values  of  K  for  the  angles  of  friction 
0°,  15°  or  30°  are  0.42,  0.39  or  0.39  giving  for  T  the  corresponding  values 
23,500,  24,500  or  24,500  pounds. 


FIG.  14. 

Allowing  for  friction  between  the  back  of  the  wall  and  the  retained  earth, 
a  close  agreement  is  again  to  be  noted  between  the  two  methods  of  computing 
the  thrust. 

2.  A  wall  with  vertical  back  20  feet  high  supports  an  embankment  as 
shown  in  Fig.  14  subject  to  a  surcharge  of  800  pounds  per  square  foot. 
Determine  the  thrust  for  the  two  conditions  of  no  surcharge  and  surcharge. 

For  the  condition  of  no  surcharge,  equation  (22)  may  be  used.  Here 
h'  —  6  feet  approximately  and  c  is  then  6/20  =  0.3.  The  angle  6  =  0°  and 
f  the  friction  between  wall  and  earth  is  ignored  (which  is  advisable  when 


38  RETAINING  WALLS 

the  back  of  the  wall  is  vertical,  as  it  is  in  this  problem)  0'  is  also  zero. 
Again  the  angle  of  repose  and  the  angle  i  are  both  equal  to  30°.  The  various 
factors  in  the  expression  then  take  the  following  values: 

L  =  I/cos2  <£  =  %.     d  =  cot   i  =  cot  <f>.     u  =  sin  </>   and   v  =  —cos  <£. 

cos  0  cot  <J> 
n   =   -    —     -  —  =  -  cot2  <£.     ra  =  sin  <£/sm  0  =  3,  and/  =  -cot2  <f> 


=  —  3.     p  =  sin  0  =  ^. 

r  =  f  2  x  |(i.3  -  ^Vi.s2  +  3  x  0.09)  2 

=  9,600  pounds. 

If  the  expression  in  (24)  had  been  used  with  K  =  yz  and  with  the  same  value 
of  c  =  0.3,  the  value  of  the  thrust  thus  found  would  be 
100  X  400  X  1.6 

2X3  '  "  1Uj7Ul 

The  latter  method,  or  rather,  equation  (24)  is  apparently  sufficiently  exact 
for  the  conditions  under  which  the  problem  was,  analyzed. 

For  the  surcharge  of  800  pounds  per  square  foot,  as  shown  in  the  figure, 
the  graphical  construction  of  Poncelet  is  employed  to  determine  the  thrust. 

Draw  aob,  making  the  triangles  aof  and  cob  of  equivalent  area.  (A  few 
trials  will  determine  the  location  of  this  line.  In  fact  the  accuracy  of  the 
problem  is  easily  satisfied  by  locating  the  line  aob  by  inspection.)  Draw 
Ab,  then  ak  parallel  to  it  and  proceed  as  before  with  this  method.  The 
thrust  is  then  the  area  of  the  thrust  triangle  inm,  multiplied  by  the  unit 
weight  of  the  earth  100  pounds  per  cubic  foot  and  is  then  equal  to 

16-7'2X10°  =  13,900  pounds. 

As  a  check  upon  this  method,  note  that  the  line  aob  makes  an  angle  of  41° 
with  the  horizontal.  The  method,  using  equation  (22)  may  be  employed 
with  the  new  surface  abi.  .  .  With  the  same  scheme  of  substitution  as 
employed  in  the  first  part  of  the  problem,  with  i  =  41°,  n  =  cot  <£  cot  i  = 
2.0  and  c  =  I%Q  =  0.7.  The  thrust  is  then  found  from  the  expression 

T  =  100X*°;X4(l.7   -   |V1.7'  +  2X0.49)  '  =  13,700 

affording  a  satisfactory  check  upon  the  graphical  calculation. 

3.  A  material  is  so  densely  compacted  and  well  drained  upon  being 
placed  behind  a  retaining  wall  that  it  is  safe  to  take  its  angle  of  slope  as  45°. 
Derive  an  expression  for  the  thrust  against  a  vertical  wall  and  also  against 
a  wall  with  a  batter  of  one  in  four. 

With  the  surface  horizontal  and  against  a  vertical  wall  the  expression 
for  K  in  both  the  Rankine  and  Coulomb  method  is 

1  —  sin  0 
1  +  sin  <t> 

which  becomes  for  a  value  of  0  =  45°,  closely  one-sixth.  The  thrust  for 
this  material  is  then  one-half  of  the  normal  thrust  against  a  vertical  wall, 
the  normal  thrust  being  that  produced  by  a  material  with  a  slope  angle  of 
30°. 


THEORY  OF  EARTH  PRESSURE  39 

The  value  of  the  slope  angle  is  14°.     From  (14)  the  expression  for  the 
thrust  becomes,  using  the  above  value  of  <j>  and  }/±  for  tan  b 


the  value  of  J  now  being  0.3,  which  may  be  compared  to  the  value  0.42  for 
<£  =  30°. 

The  corresponding  values  for  the  thrust  as  determined  by  the  method  of 
the  sliding  wedge  are  easily  found  by  proper 
substitution  of  the  value  of  <£  =   45°  in  the 
constant  K,  in  the  expression  as  given  in  (25). 
This  arithmetic  work  need  not  be  given  here. 

4.  A  building  wall  running  parallel  to  a  re- 
taining wall,  as  shown  in  Fig.  15  carries  a  load 
of  one  ton  per  square  foot  and  has  a  spread  of 
four  feet  at  a  base  four  feet  below  the  top  of  -piG  15 

the   retaining   wall.      The   retaining    wall   is 

subject  to  no  surcharge  load  other  than  that  produced  by  the  bearing  wall. 
What  is  the  total  thrust  upon  the  wall  and  where  is  it  located? 

Referring  to  Fig.  15,  the  value  of  L/a  is  four  tons  or  8000  pounds  per  lineal 
foot  of  wall.     There  is  no  surcharge  and  with  h  =  25  feet 

«)'  =  MX626  _  31;250pound, 


The  ratio  L/a  to  gh*(l  +  c)2/2  is  0.256.  This  is  the  value  of  the  ratio  r. 
With  this  value  entering  curve  plate  No.  1,  the  value  of  m  for  a  maximum 
wedge  of  sliding  is  74°.  It  is  observed  that  this  plane  will  intersect  the  foot- 
ing and  accordingly  the  maximum  plane  of  slip  is  made  to  pass  through  the 
inner  edge  of  the  base.  This  gives  a  value  of  69°  for  m. 
The  thrust  due  to  the  concentrated  load  is 

8000  tan(69°  -  30°)  =  6480  pounds. 
That  due  to  the  earth  wedge  is 

100  *  625  cot  69°  tan(69°  -  30°)  =  9700  pounds. 

The  point  of  application  of  the  thrust  due  to  the  concentrated  load  is  10.5 
feet  above  the  base  of  the  vertical  wall.  That  of  the  earth  wedge  is  one- 
third  of  the  distance  up  or  8.33  feet.  The  total  thrust  is  then  6500  + 
9700  =  16,200  pounds  and  is  located 

6500  X  10.5  +  9700  X  8.33 


6500  +  9700 


=  9.2  feet  above  the  base  of  the  wall. 


Assuming  that  the  transmitted  pressure  of  the  bearing  wall  is  contained 
within  planes  making  an  angle  of  30°  with  the  vertical,  at  a  point  approxi- 
mately 11  feet  below  the  surface  the  distribution  of  the  load  would  strike 
the  back  of  the  retaining  wall.  With  a  uniform  distribution  of  the  load  at 
this  plane,  the  intensity  of  the  transmitted  pressure  is  800%2  =  670 
pounds  per  square  foot.  If  this  is  treated  as  a  surcharge  at  the  surface  and 


40  RETAINING  WALLS 

equation    (24)   is  employed  to  obtain  the  thrust,  c  is  then  6-Ks   =  0-27. 

With  K  taken  as  >| 

„,       100X625X1.54 

T  =  -         2X3  --  =  16'05°Pounds- 

1  81  X  25 
From  Table  3  the  point  of  application  of  this  thrust  is  located  -^—  - 

O    /\    J-.OTT 

=  9.8  feet  above  the  base  of  the  wall.  See  page  30  for  a  discussion  of 
the  use  of  this  method  of  analysis  as  a  check  upon  the  prev  ous  method. 

As  a  problem  illustrative  of  /the  action  of  saturated  earth  the 
author  of  the  paper  on  page  32  has  given  the  following  example:1 

"Take  for  example  a  wall  supporting  ten  feet  of  earth  the  lower  6  ft. 
of  which  are  below  water  level  and  hence  saturated.  Assume  dry 
earth  at  100  pounds  per  cubic  foot  and  earth  under  water  at  70  pounds 
per  cubic  foot.  Assume  a  natural  slope  for  dry  earth  of  1.5  to  1 
(0i  =  33°41')  and  for  earth  under  the  water  of  2.5  to  1  (02  =  21°48'). 

"Lateral  pressure  at  the  plane  of  saturation  due  to  dry  earth  =  100 
X  4  X  (1  —  sin  0i)/(l  +  sin  0J  =  114.4  Ibs.  per  square  foot. 

"Lateral  pressure  at  the  plane  of  saturation  due  to  saturated  earth  = 

100  X  4  X  -j~  ^  =  183.2  Ibs.  per  square  foot. 
"Lateral  earth  pressure  at  the  bottom 

(100  +  4  +  70  X  6)j"Sm^2  =  374.6  Ibs.  per  sq.  ft. 
i  "T"  sin  <p2 

"Hydrostatic  pressure  at  the  bottom  =  62.5  X  6  =  375  Ibs.  per 
square  foot. 

"Total  lateral  pressure  at  the  bottom  =  374.6  -f  375  =  749.6  Ib.  per 
sq.  ft. 

"Total  resultant  lateral  pressure  above  the  plane  of  saturation  per 
foot  length  of  wall  is  114.4  X  0.5  X  4  =  228.8  Ib.  This  is  applied  at  a 
point  l^j  ft.  from  the  plane  of  saturation  or  7>£  ft.  from  the  bottom  of 
the  wall. 

"Total  resultant  lateral  pressure  below  the  plane  of  saturation  is  0.5 
(183.2  +  749.6)  X  6  =  2798.4  Ib.  This  is  applied  at  a  distance  of 
6(749.6  +  2  X  183.2) 

3(749.6  +  183.2)     Or  2A  feet  from  the  bott°m- 

"The  resultant  lateral  pressure  against  the  wall  per  foot  of  length 
is  then  228.8  +  2798.4  =  3027.2  Ib.  This  is  applied  at  a  distance  of 
228.8  X  7.3  +  2798.4  XA  bottom  „ 


BIBLIOGRAPHY 

For  an  exhaustive  bibliography  on  the  various  theories  and  experiments 
upon  earth  pressures,  both  active  and  passive  see  HOWE,  "  Retaining  Walls," 
5th  Ed.  (see  also  Appendix)  . 

1  A.  G.  HUSTED,  Engineering  News-Record,  Vol.  81,  p.  442. 


THEORY  OF  EARTH  PRESSURE  41 

The  following  is  a  list  of  interesting  papers  upon  the  subject  matter  of 

the  chapter. 

Earth  Pressures:  A  practical  comparison  of  theory  and  experiments, 
CORNISH,  Trans.  A.  S.  C.  E.,  Ixxxi,  p.  191. 

Cohesion  in  Earth:  CAIN,  Trans.  A.  S.  C.  E.,  Ixxx,  p.  1315. 

Earth  Pressure  Lateral:  Cornell  Civil  Engineer,  April,  1913. 

Lateral  Pressure  of  Clay:  W.  L.  COOMBS,  Journal  Western  Society  of  Engi- 
neers, Vol.  17,  p.  746. 

Retaining  Wall  Theories:  PERRY,  Journal  Western  Society  of  Engineers, 
Vol.  19,  p.  113. 

Retaining  Walls:  Based  entirely  upon  the  theory  of  friction,  P.  DOZAL, 
Buenos  Aires.  Translated. 


CHAPTER  II 


DESIGN  OF  GRAVITY  WALLS 

Location  and  Height  of  Wall. — The  need  for  a  retaining  wall 
arises  from  the  construction  of  a  cut  or  an  embankment,  whose 
side  banks  are  not  permitted  to  take  their  natural  slopes.  Where 
the  amount  of  land  necessary  for  the  construction  of  such  a  fill 

or  cut  is,  to  all  intents,  un- 
limited, the  wall  may  be 
located  at  any  point  where 
economy  dictates  that  a  wall 
of  the  necessary  height  and 
section  is  cheaper  than  the 
additional  cut  or  fill  which  it 
FlG  16  replaces.  Thus  in  Fig.  16  the 

wall    replaces  all  fill  shown 

cross-hatched.  A  comparative  estimate,  taking  into  considera- 
tion the  cost  of  masonry,  of  embankment,  or  excavation  for  the 
wall  footing,  will  show,  after  a  few  trials  as  to  location,  at  what 
point  the  wall  should  be  placed  to  obtain  the  minimum  cost. 

If  the  wall,  however,  is  to  run  along  a  highway  or  other  fixed 
property  line,  then,  this  at  once  determines  its  location.  Again, 


Easement 
Line  .-- 


RoadSurface 


FIG.  17. 


FIG.  18 


in  railroad  work  through  cities,  especially  grade  elimination  and 
track  elevation  work,  easements  are  costly  and  are  generally  re- 
stricted by  the  municipalities  which  grant  them,  so  that  it  is 
necessary  to  get  the  wall  as  close  to  the  tracks  as  possible,  whence 
a  wall  is  placed  as  shown  in  Fig.  17.  Even  in  the  case  where  ease- 

42 


DESIGN  OF  GRAVITY  WALLS  43 

ments  are  cheap  and  unlimited,  an  eye  to  future  development  and 
consequent  increased  trackage  may  make  it  desirable  to  so  con- 
struct a  wall,  that  the  additional  fill  necessary  for  the  future  tracks 
may  easily  be  placed.  In  Fig.  18  the  wall  may  be  so  built,  that, 
with  placing  a  new  top  above  A,  the  section  will  be  ample  to  take 
care  of  the  new  fill  and  live  load,  or  the  wall  may  be  built  to  the 
future  required  height  at  once.  This  latter  may,  however,  prove 
unsightly. 

General  Outlines  of  the  Wall. — The  section  of  a  wall  should  be 
so  chosen  that,  at  a  minimum  cost,  it  yields  a  maximum  area  for 
the  improvement  work.  When  this  w^ork  runs  through  valuable 
property  acquired  at  high  cost,  so  that  every  square  foot  possible 
must  be  made  available  for  the  roadway  or  tracks,  the  front 
face,  on  the  property  line,  should  be  made  vertical  as  shown  in 
Fig.  17  and  placed  as  close  to  the  line  as  the  details  of  the  coping 
and  footing  will  permit.  To  insure  no  possible  encroachment  at 
a  future  date,  due  to  settlement  of  the  wall,  surveying  or  con- 
struction errors  and  the  like,  it  is  better  to  place  the  coping  a 
few  inches  back  from  the  line.  The  coping  usually  projects  a 
few  inches  beyond  the  face  of  the  wall. 

Before  entering  into  a  discussion  of  the  relative  merits  of  walls 
with  various  outlines,  it  is  necessary  that  the  principles  upon 
which  the  walls  are  designed,  be  first  explained.  This  will  be 
done  in  the  following  pages.  The  section  of  the  wall  may  be 
controlled  not  only  by  these  general  principles,  but  also  by  specific 
limitations  foreign  to  the  actual  stress  system  existing  in  the  wall. 
Architectural  treatment  may  determine  the  shape  of  the  wall, 
when  the  wall  is  part  of  some  general  landscape  scheme.  The 
selection  of  a  type  of  wall  that  will  suit  peculiar  foundation  condi- 
tions is  discussed  in  detail  in  later  chapters.  Generally  speaking, 
however,  that  section  of  wall  is  chosen  which  can  be  most  econom- 
ically and  expeditiously  built. 

The  Two  Classes  of  Retaining  Walls. — 'Retaining  walls  fall  into 
two  broad  classes.  The  walls  which  retain  an  earth  bank  wholly 
by  their  own  weight  are  termed  gravity  walls.  This  type  is  dis- 
cussed in  the  present  chapter.  Those  which  utilize  the  weight 
of  the  earth  bank  in  sustaining  the  pressures  of  the  bank  form  the 
reinforced  concrete  type  of  walls.  This  latter  class,  because  of  the 
mobile  character  of  reinforced  concrete  has  an  infinite  variety 
of  shapes.  The  following  chapters  will  take  up  in  detail  the 
analysis  of  the  shapes  occurring  in  ordinary  construction  work. 


44 


RETAINING  WALLS 


Since  the  active  element  of  support  in  the  gravity  wall  is  the 
material  out  of  which  it  is  composed,  the  wall  may  be  made  of 
other  materials  besides  concrete.  The  reinforced  walls  are  made 
of  concrete  and  steel. 

Fundamental  Principles  of  Design.— A  retaining  wall,  in  sup- 
porting an  earth  bank  must  successfully  withstand  the  following 
possible  modes  of  failure : 

(a)  The  overturning  moment  caused  by  the  earth  thrust  may 
exceed  the  stability  moment  of  the  weight  of  the  wall,  or  in  the 
case  of  the  cantilever  type,  of  the  combined,  weight  of  the  wall 
and  relieving  earth  weights.  Thus  in  Fig.  19  the  thrust  moment 
Tl  is  greater  than  the  stability  moment  Gg,  and  the  wall  will 


FIG.  19. — Criterion  of  overturning. 


16 

FIG.  20. — Criterion  of  sliding. 


rotate  about  its  toe.  To  remedy  this,  the  weight  G  or  the  lever 
arm  g  is  increased  by  adding  to  the  dimensions  of  the  wall,  usually 
by  widening  the  base. 

(6)  The  pressure  on  the  toe  caused  by  the  resultant  forces  of 
the  thrust  and  weight  of  wall  and  earth  may  exceed  the  bearing 
power  of  the  soil  at  that  point,  crushing  the  ground  and  causing 
the  wall  to  tilt  forward  and,  in  the  extreme  case,  topple  over. 
The  remedy  lies  in  a  wall  properly  shaped  and  dimensioned  to 
insure  safe  soil  pressures,  or  where  dimensions  alone  will  not 
suffice  the  preparation  of  a  proper  foundation  either  by  further 
excavation  to  a  better  bottom  or  by  the  use  of  timber  or  pile 
foundations. 

(c)  The  frictional  resistance  between  the  wall  base  and  the 
foundation  may  be  insufficient  to  overcome  the  horizontal  com- 
ponent of  the  thrust  and  the  wall  will  slide  forward  along  the  base. 
In  Fig.  20  fG  is  less  than  TV  /  is  the  coefficient  of  friction,  a 
table  of  which  for  various  materials,  is  shown  here  (Table  6). 
Th  is  the  horizontal  component  of  the  thrust.  With  a  wall  pro- 
perly proportioned  against  failure  through  overturning  or  exces- 


DESIGN  OF  GRAVITY  WALLS  45 

sive  bearing  on  the  foundation,  this  condition  rarely  exists.  It 
is  most  likely  to  occur  on  a  clay  bottom,  if  water  is  present,  since 
the  wet  clay  acts  as  a  lubricant.  To  remedy  a  condition  of  this 
kind,  the  base  may  either  be  widened,  increasing  the  weight  on 
the  wall,  or  a  bottom  may  be  prepared  offering  mechanical  as  well 
as  frictional  resistance  to  sliding.  If  narrow  trenches  are  dug  in 
the  foundation,  projections  will  be  formed  which  will  materially 
increase  the  resistance.  Again,  the  bottom  may  be  tilted  up- 
wards towards  the  toe,  giving  a  horizontal  component  of  resis- 


FIG.  21. — Types  of  bottoms  to  increase  resistance  against  sliding. 

tance  in  addition  to  the  frictional  (see  Fig.  21  for  both  cases). 
Filling  the  foundation  trench  completely  with  masonry,  so  that 
the  front  of  the  wall  butts  against  the  original  earth  of  the  trench 
(not  any  backfill)  may  also  prove  efficacious. 

TABLE  6 


Character  of  foundation 

Coefficient 

Dry  clay  .  . 

.50 

Wet  or  moist  clay.  .                                                       .    . 

33 

Sand 

40 

Gravel  .                                    .    .          .... 

60 

Wood  (with  grain) 

60 

Wood  (against  grain)  

.50 

These  are,  then,  the  potential  modes  of  failure  of  a  retaining 
wall,  and  the  wall  satisfying  most  economically  these  criteria 
against  failure  has  been  properly  designed. 

To  recapitulate,  the  following  equations  must  be  satisfied: 

(a)  Gg  must  be  greater  than  Tt. 

(b)  Si  must  be  less  than  S  (where  Si  is  the  toe  pressure  actually 

induced  and  S  is  the  permissible  soil  pressure.) 

(c)  fG  must  be  greater  than  Th. 

Concrete  or  Stone  Walls. — -In  spite  of  the  well-nigh  universal 
adoption  of  concrete  as  a  retaining  wall  material,  many  yards  of 


46  RETAINING  WALLS 

stone  wall  are  still  being  built.  Under  certain  conditions,  this 
type  of  wall  is  the  more  economical  one.  The  cut  stone  walls, 
however,  with  their  ashlar  or  coursed  masonry  faces  are  much 
more  costly  than  the  concrete  walls  and  are  only  used  when 
necessitated  by  architectural  treatment.  With  the  development 
of  the  artistic  treatment  of  concrete  faces  and  with  the  ability  to 
duplicate  practically  every  cut-stone  effect  in  concrete,  the  need 
of  stone  walls  for  even  this  purpose  is  rapidly  diminishing.  The 
rubble  walls,  both  mortar  and  dry,  do  have  an  important  applica- 
tion and  where  local  stone  cuts  are  available,  are  far  the  cheapest 
material  out  of  which  to  build  the  wall. 

When  a  wall  is  to  be  built  adjacent  to  property,  to  which  no 
access  is  permissible,  even  during  construction,  thus  preventing 
the  placing  of  the  bracing  and  concrete  forms,  a  stone  wall  be- 
comes a  very  convenient  type  of  wall  to  build.  Rubble  walls 
were  so  used  in  the  track  elevation  of  the  Philadelphia,  German- 
town,  and  Norristown  Railroad  through  Philadelphia.1 

The  dry  rubble  wall  is  frankly  a  temporary  expedient,  awaiting 
further  local  improvements,  upon  the  arrival  of  which,  the  need 
for  the  wall  itself  is  either  removed  or  else  the  walls  are  replaced 
by  those  of  more  permanent  and  pleasing  effect.  The  word 
"  temporary "  should  be  used  most  qualifiedly,  for  many  dry 
rubble  walls  have  existed  for  long  periods  of  time,  exceeding,  by 
far  their  expected  duration  of  life.  In  municipal  improvements, 
as  for,  example  the  grading  of  a  highway,  leaving  surrounding 
unimproved  property  below  the  future  grade,  it  is  customary  to 
place  a  dry  rubble  wall  along  the  highway  with  the  expectation 
that  when  the  adjacent  property  is  improved  or  graded,  the  wall 
will  either  be  removed  or  buried  (see  Plate  1,  Fig.  la). 

The  cement  rubble  wall  is  of  as  permanent  a  nature  as  the 
concrete  wall.  Its  face,  unless  more  or  less  screened  is  not  as 
pleasing  as  a  concrete  face  when  viewed  at  close  range.  At  com- 
paratively small  distances  away,  however,  it  presents  quite  a 
pleasing  effect,  the  variegated  coloring  of  the  local  stone  showing 
to  advantage  (see  Plate  1,  Fig.  16). 

The  stone  walls  require  a  distinct  class  of  labor,  familiar  with 
the  work.  Stone  masons  are  not  always  available  and  because 
of  the  diminishing  amounts  of  stone  walls  built,  are  becoming 
fewer  in  number.  The  universal  adaptability  of  concrete,  its 
independence  of  local  material  conditions  and  the  large  amount 

1  See  S.  T.  WAGNER,  Trans.  A.S.C.E.,  Vol.  Ixxvi. 


PLATE  I 


Fia.  A. — Dry  rubble  wall  along  highway. 


FIG.  B. — Characteristic  appearance  of  cement  rubble  wall. 

(Facing  page  46) 


DESIGN  OF  GRAVITY  WALLS  47 

of  concrete  laborers  and  foremen  all  tend  to  explain  the  waning 
popularity  of  stone  masonry.1 

Where  the  selection  of  the  material  out  of  which  the  wall  is  to 
be  built  is  governed  solely  by  economic  reasons,  then,  with  labor 
and  material  conditions  of  equal  weight  the  costs  of  the  dry 
rubble  wall,  the  cement  rubble  wall  and  the  concrete  wall  stand 
in  the  order  one,  two  and  three,  i.e.,  the  cost  of  the  cement  rubble 
wall  is  twice  that  of  the  dry  rubble  wall  and  the  concrete  wall 
three  times  that  of  the  dry  rubble  wall.  It  is  understood  that 
there  are  available  local  stone  quarries  for  the  rubble  wall. 
A  very  long  haul  for  the  stone  makes  the  cost  of  the  wall  far  too 
high  to  permit  a  serious  consideration  of  its  construction. 

When  using  a  dry  wall,  care  must  be  taken  to  allow  for  the 
voids  in  assuming  the  weight  of  the  masonry.  The  voids  may 
vary  from  15  to  40  per  cent,  of  the  section.  A  problem  at  the 
end  of  this  chapter  brings  out  this  in  some  detail. 

Thrust  and  Stability  Moments.  —  The  method  of  determining 
the  thrust  upon  the  back  of  a  gravity  wall  follows  the  recom- 
mended form  of  procedure  given  on  page  16.  The  thrust  T  upon 
the  back  of  the  wall  is  located  at  a  point  Bh  above  the  bottom  of 
the  wall,  where  the  value  of  B  is  found  from  Table  3.  The  stand- 
ard type  of  surcharge  loading  of  height  In!  is  used  (see  Fig.  5)  and 
the  ratio  In!  '  /h  is  denoted  by  c.  The  amount  of  the  thrust  is 


where  J  is  the  adopted  earth  pressure  coefficient  to  be  taken  from 
equation  (14)  or  from  Table  1.  The  unit  weight  of  earth  is  g 
(replacing  w  in  the  original  equation  to  avoid  confusion  with  a 
more  natural  form  of  lettering  used  in  the  following  algebraic 
work)  . 

If,  under  special  conditions  (see  problems  at  the  end  of  this 
chapter)  it  is  decided  to  use  the  method  of  the  maximum  wedge 
of  sliding,  with  the  equation  24  on  page  15,  the  thrust  is 


where  K  is  the  earth  pressure  coefficient  of  this  method  corre- 
sponding to  J  above  and  is  to  be  taken  from  equation  (24)  or 

1  See  Engineering  News-Record,  Vol.  81,  p.  890  for  a  description  of  the 
use  of  dry  rubble  walls  to  retain  the  Hetch-Hetchy  Railroad.  The  cuts 
for  the  highway  afforded  large  amounts  of  stone. 


48 


RETAINING  WALLS 


from  Table  2.  Unless  the  back  of  the  wall  has  a  small  batter 
(less  than  5°)  it  is  recommended  that  a  value  of  0'  =  30°  be  used 
in  finding  the  value  of  K. 

Following  are  some  general  relations  between  the  wall'  factors 

and  the  thrust,  covering  all  shapes  of 
gravity  walls  and  all  varieties  of  earth 
pressures. 

Let  Fig.  22  represent  a  general  sec- 
tion of  gravity  wall.  Assume  that 
the  thrust  has  been  found,  in  value  T 
and  located  at  a  point  Bh  vertically 
above  the  base.  The  weight  of  the 
wall  G  is  usually  found  by  breaking 
up  the  figure  as  shown  into  triangles 
and  rectangles.  Algebraically  then, 
by  taking  moments  about  some  con- 
venient point,  as,  for  example,  at  the  toe  A,  both  the  thrust 
moment  Tt  and  the  stability  moment  G±g\  +  G2g2  +  G3g3 
are  easily  found.  Graphically  by  means  of  an  equilibrium 
polygon  it  is  a  simple  matter  to  locate  the  resultant  of  the  forces 
both  in  amount  and  in  point  of  application.  In  the  above  alge- 
braic method  it  is  necessary  to  proceed  further  to  obtain  the 
resultant  in  both  location  and  in  amount.  Fig.  23  shows  the 


FIG.  22. — Stress  system  in 
gravity  wall. 


Intersection 
of  fays  1+5 


,  Drawn  Parallel 
'to  R  in  Polygon 

FIG.  23. — Graphical  analysis  of  gravity  wall  stresses. 


method  of  applying  the  thrust  polygon  to  the  determination  of 
the  stability  of  the  wall. 

The  wall  is  on  the  verge  of  overturning  when  the  stability 
moment  is  equal  to  the  thrust  moment  or  what  is  the  same  thing 
when  the  resultant  just  intersects  the  toe  of  the  wall.  For  this 
condition  the  factor  of  safety  is  one. 


DESIGN  OF  GRAVITY  WALLS  49 

As  long  as  the  stability  moment  exceeds  the  thrust  moment, 
or  as  long  as  the  point  of  application  of  the  resultant  falls  within 
the  base,  the  wall  is  safe  against  overturning.  The  proper 
location  of  the  resultant  depends  not  only  upon  the  factor  of 
safety  thought  desirable  but  also  upon  the  question  of  a  satis- 
factory foundation  pressure.  Before  entering  upon  a  discussion 
of  a  safety  factor  against  overturning,  it  may  be  well  to  discuss 
the  matter  of  foundations. 

Foundations,  those  most  vexing  problems  of  engineering 
practice,  are  of  paramount  importance  in  both  wall  design 
and  construction.  Generally  a  correct  foundation  design  de- 
mands a  uniform  distribution  of  load  as  its  most  important 
premise.  Unfortunately,  the  economics  of  retaining  walls 
usually  forbid  the  fulfillment  of  this  premise.  The  wall  is 
considered  satisfactorily  designed  so  long  as  the  resultant  of  the 
pressure  on  the  base  falls  within  the  middle  third  of  the  base, 
and  more  often  at  the  outer  edge  of  this  middle  third,  so  that 
the  pressure  intensity  on  the  base  varies  from  nothing  at  the 
heel  to  the  maximum  at  the  toe. 

For  foundations  varying  from  rock  ,to  hard  soils,  such  as 
coarse  sands  and  gravels  or  loamy  soils,  i.e.,  a  mixture  of  gravelly 
sand  and  clay,  the  relative  settlements  due  to  the  varying  loads 
is  small  and  a  non-uniformly  distributed  load  may  safely  be 
placed  upon  them.  For  the  finer  sands,  wet  soils,  reaching  down 
to  the  plastic  bottoms,  it  is  imperative  to  have  a  uniform  dis- 
tribution of  pressure  and  foundations  must  be  designed  to 
secure  this  or  recourse  must  be  had  to  special  types  of  walls, 
such  as  the  cellular  and  similar  types  (see  later  pages). 

There  is  no  intention  of  entering  into  a  detailed  analysis  of 
the  proper  selection  and  preparation  of  a  foundation.1  A  brief 
description  only  of  the  various  types  of  bottoms  will  be  given. 
Various  phases  of  this  subject,  however,  will  be  taken  up  under 
the  headings  of  "  Varied  Types  of  Walls,"  "  Settlement,"  etc. 

Rock  is  an  elastic  term,  embracing  all  the  types  from  a  dis- 
integrated product,  that  can  easily  be  picked  and  shovelled 
to  the  hard  gneiss,  trap  and  granite  which  prove  so  costly  to 
drill  bits.  The  poor  rocks,  when  stripped  of  a  one  or  two  foot 
layer  usually  present  a  bottom  sufficiently  strong  to  take  as  heavy 
a  load  as  the  safe  crushing  strength  of  the  wall  material  will 
permit,  and  this  is,  of  course,  the  maximum  pressure  that  can 

1  See  texts  by  JACOBY  &  DAVIS;  PATTON;  FOLWELL,  etc. 


50  RETAINING  WALLS 

be  allowed  on  any  masonry  foundation.  Under  these  conditions, 
the  resultant  may  intersect  the  outer  edge  of  the  middle  third 
with  a  triangular  distribution  of  base  loading.  Occasionally  the 
resultant  is  permitted  to  fall  outside  the  middle  third,  so  that  the 
wall  bears  on  only  part  of  the  foundation.  While,  theoretically, 
tension  must  then  exist  between  the  base  and  the  foundation  to- 
wards the  heel  of  the  wall,  the  rock  is  unyielding,  so  that  there 
can  be  no  opening  at  the  heel  while  the  criteria  of  overturning 
and  safe  bearing  loads  are  satisfied.  In  the  gravity  walls,  when 
this  type  of  foundation  is  adopted,  care  must  be  taken  that  the 
tension  then  developed  in  the  back  of  the  wall  at  the  base  does 
not  exceed  the  tensile  strength  of  the  masonry.  If  it  does,  it  is 
necessary  to  reinforce  the  back  with  rods. 

With  a  rock  bottom  well  cleaned,  left  in  the  usual  rough 
condition,  and,  with  a  good  bond  secured  between  it  and  the  base 
of  the  wall,  there  is  ample  resistance  to  sliding. 

Shales,  cementatious  gravels,  coarse  sand  and  gravel,  in  similar 
fashion  present  but  little  difficulty  and  it  is  customary,  here  also, 
to  permit  a  triangular  distribution  of  soil  pressure.  Shading 
off  into  the  finer  sands,  dry  clays  and  bottoms  of  like  type  with 
moderately  yielding  propensities,  a  theoretical  discussion1  of 
passive  earth  pressures  seems  to  indicate  that  in  yielding  soils 
there  is  an  upward  heaving  of  the  soil  adjacent  to  the  down- 
ward loads,  so  that,  to  counteract  this  tendency,  there  must 
be  a  minimum  downward  pressure  on  the  base.  For  this 
reason,  the  resultant  of  the  pressures  should  strike  the  base 
within  the  middle  third,  giving  a  trapezoidal  distribution  of 
pressure. 

Coming  down  to  the  plastic  bottoms,  there  must  be  a  uniform 
distribution  along  the  base  not  to  exceed  the  safe  bearing  value 
of  the  soil  in  question.  If  this  is  not  possible  it  is  necessary  to 
place  piles.  It  is  highly  desirable  that  the  piles  carry  equal 
loads.  If  the  base  pressure  is  not  uniform  a  uniform  pile 
loading  may,  nevertheless,  be  secured,  by  proper  spacing  of 
the  piles. 

Distribution  of  Base  Pressures. — The  analysis  of  the  loadings 
upon  the  wall  determines,  finally,  the  location  and  amount  of  the 
resultant  pressure  upon  the  base  of  the  wall.  Since  this  re- 
sultant force  is  eccentrically  placed  upon  the  base,  it  is  necessary 
to  obtain  the  manner  of  the  distribution  of  the  pressure  due  to 

1  HOWE,  "Retaining  Walls,  Earth  Pressures  and  Foundations." 


DESIGN  OF  GRAVITY  WALLS 


51 


this  resultant.  The  vertical  component  of  the  resultant  is  ana- 
lyzed here;  the  horizontal  component  affecting  only  the  frictional 
resistance  between  the  wall  and  the  earth. 

Referring  to  Fig.  24,  let  R  be  the  vertical  component  of  the 
resultant  of  all  the  pressures  upon  the  base.  $1  and  82  are  the 
extreme  pressure  at  the  toe  and  heel 
respectively.  With  these  limiting  in- 
tensities found  all  the  necessary  data 
for  the  footing  is  had. 

Take  moments  about  (the  heel) 


and 


FIG.  24. — Foundation 
pressures. 


Si  +  2S2  =  QkR/w 


(37) 


Again,  since  the  area  of  the  trapezoid  is  equivalent  to  the  value 
of  the  resultant  R 


Si  +  S2  =  2R/w 
Solving  these  simultaneous  equations,  there  is 


(38) 


Si  = 


(39) 


(40) 


When  k  =  Hj  i.e.,  when  the  resultant  intersects  at  the  outer 
edge  of  the  middle  third  —  a  very  common  condition,  Si  =  2R/w 
and  82  =  0.  When  k  =  M>  *•«•,  when  there  is  a  uniform 
distribution  of  pressure  along  the  base  Si  =  82  =  R/w. 

Note  that  when,  k  is  less  than  one-third,  there  is  pressure  along 
only  a  portion  of  the  base.     The  point  of  zero  intensity  is  given  by 


x  =T: 


w  1  -  3k 
3  1  -  2k 


(41) 


where  x  is  the  distance  from  the  heel  to  the  point  of  zero  in- 
tensity. 

Table  7  gives  the  permissible  intensities  of  soil  pressures  as 
allowed  by  the  various  codes. 


52  RETAINING  WALLS 

TABLE  7. — PERMISSIBLE  SOIL  PRESSURES  IN  TONS  PER  SQUARE  FOOT 


Soil 

A 

B 

c 

D 

E 

Quicksand  silt 

y>-\ 

I 

Clay,  soft  

K-2 

2 

1 

I 

1 

Clay  and  sand  

2-4 

2 

2 

+2 

Sand  clean  dry 

2-4 

4 

3 

3 

Sand  compacted,  well  cemented  . 
Gravel  and  coarse  sand  

4-6 
6-8 

6 

6 

4 

6 

Gravel  and  coarse  sand  well  com- 
pacted      .      

8-10 

10 

10 

Clay,  hard,  moderately  dry  
Clay  hard  dry 

4-6 
6-8 

4 

4 

4 

Rock,  soft  to  bard  

5-200 

75* 

8-40 

12-20 

S, 


-kw 


A.  Prof.  Cain. 

B.  Public  Service  Commission,  1st  District,  New  York  City. 

C.  Building  Code,  New  York  City. 

D.  Building  Code,  Dist.  of  Washington. 

E.  Building  Code,  Baltimore. 
*  Sound  ledge  rock. 

f  Clay  or  clay  mixed  with  sand,  firm  and  dry.     3  tons. 

Proper  Centering  for  Piles. — Since  the  retaining  wall  brings 
a  non-uniform  distribution  of  loading  upon  the  base,  a  uniform 
spacing  of  piles  would  produce  unequal  loading  upon  them. 
This  is  not  a  desirable  type  of  loading 
for  piles.  The  following  is  a  method 
of  so  spacing  the  piles  as  to  secure  a 
uniform  loading. 

The  piles  may  be  spaced  either  in 
rows  parallel  to  the  face  of  the  wall,  or 
in  rows  perpendicular  to  the  face  of 
the  wall.  A  graphic  and  an  analytic 
method  are  outlined  below  for  either 
of  these  two  methods  of  spacing  the  piles. 

Let  P  be  the  safe  bearing  value  per  pile.  In  Fig.  25  divide 
the  base  into  a  series  of  strips  of  equal  width  v.  From  the 
eccentric  position  of  R  determine  the  extreme  bearings,  Si  and  $2 
and  lay  these  off  to  scale.  The  soil  pressure  in  any  strip  v,  SV) 
is  readily  obtained  by  scaling  the  figure.  vSv  then  gives  the  total 
load  on  the  v  strip  taken  for  a  unit  width  of  wall.  Dividing 
P  by  this  product  determines  the  spacing  necessary  in  that  strip. 
The  minimum  spacing  of  piles  is  about  three  feet,  so  that,  when 


FIG.  25. — Pile  spacing. 
Case  I. 


DESIGN  OF  GRAVITY  WALLS 


53 


the  spacing  in  a  strip  is  found  to  be  less  than  this  minimum,  it 
is  necessary  to  take  the  strips  closer  together.  When  this  fails 
the  base  must  be  widened  by  placing  a  toe  extension. 

The  piles  may  be  spaced  perpendicularly  to  the  paper  at  equal 
intervals,  but  at  varying  distances  along  the  base  of  the  wall 
(see  Fig.  26) .     Assume  that  a  width 
of  wall  is  taken  (perpendicular  to 
the  sheet)  equal  to  the  permissible 
or  desirable  spacing  of  piles.     The 
values  of  R,  Si  and  S2,  as  found 
above   are   increased  accordingly. 
Making  a  scale  layout  as  above, 
trial    irregular    widths   are  taken 
decreasing  in  width  towards  the  FlG.  26.— Pile  spacing.    Case?!, 
toe,  each  being  equivalent  to  the 

safe  bearing  of  one  pile.  The  following  is  an  analytic  discussion 
of  the  two  cases. 

Case  I. — From  the  geometry  of  Fig.  25  the  total  pressure  in 
any  width  v  of  the  base  (a  unit's  thickness  of  wall  is  assumed)  is 


vS, 


w 


(Si  -  S*) 


i  is  the  number  of  the  division,  counting  from  the  back  of  the 
wall. 

Replacing  Si  and  82  by  their  values  in  terms  of  R  and  k 

.  £       S,(  =  ~  (3fc  -  1 


3 


(42) 


Since  the  pile  can  take  P  as  a  safe  load,  the  required  spacing  of 
piles  in  the  "i"  the  row  is,  then 


Case  II.  —  Let  it  be  assumed  that  the  rows  of  piles,  parallel  to 
the  page,  are  spaced  m  feet  apart.  The  total  vertical  load  on  the 
foundation  is  then  mR  and  if,  as  above,  P  is  the  safe  load  per  pile, 
the  number  of  piles  required  in  each  thickness  m  of  the  wall  is 
mR/P  =  n  and  this  is  the  required  number  of  spaces  of  equal  area 
into  which  it  is  required  to  divide  the  trapezoid,  in  Fig.  26.  Com- 
plete the  triangle  as  shown  and  let  the  area  of  SCO  be  P0.  The 
area  of  any  other  triangle,  bound,  say,  by  the  vertical  side  b<, 


54  RETAINING  WALLS 

as  base,  is  P0  +  iP,  where  i  is  the  number  of  divisions,  or  of 
piles,  from  the  back  of  the  wall.  Since  the  areas  of  similar 
triangles  are  to  each  other  as  the  square  of  their  homologous 
sides 

6;2  Po  +  iP  6i2      Po  +  P   c 

p— ;  -  Po  +  (i  -  i)P>  then  v  =    -p— ,  Similarly  V  =  W 


Extending  this  result  to  the  general  case 

A^  (44) 

Let  Z;  be  the  distance  from  B  to  the  corresponding  $  line 

then  It  =  64  -  bQ  =  60  U/P°  JJ"  ^   -  l)  flwce  P0  =^|-2,  60  = 

\  \       -to  / 


and  if,  finally,  Si  and  A^2  are  replaced  by  their  values  in  terms  of 
R  and  k 


That  the  distance  between  the  two  piles  adjacent  to  the  toe 
shall  not  be  less  than  a  specified  amount  a  (usually  about  three 
feet)  it  may  be  necessary  to  extend  the  base  by  means  of  a  toe 
With  sufficient  exactness  the  distance  a  may  be  taken  as  one-half 
the  distance  between  the  toe  and  the  point  /n_2.  Then 

w0  —  ln-2  =  2a 

Replacing  ln-z  by  its  value  from  (45),  simplifying  the  resulting 
equation  and  eliminating  the  radical  and  putting  2a/wQ  =  X 


3k-  1  F  n 

and  solving  for  k 


6X(1  -  X) 

If  the  width  including  the  toe  extension  is  WQ,  and  the  width  with- 
out the  toe  extension  is  w,  letting  2a/w  =  X'  and  noting  that 

WQ  =  W(!+  i)  and  X'  =  X  (1  +  *)  also  k  =  ——    (see    Fig.    24). 


DESIGN  OF  GRAVITY  WALLS 


55 


Equation  (46)  becomes  a  cubic  in  (1  +  i)  or  u,  A  =  3X'  [X'  + 
2(1  -e)],B  =6X'2(1  -M); 

-  w3  +  2X  V  -  Au  +  B  =  0.  (47) 

In  view  of  the  fact  that  i  is  small  in  comparison  with  unity,  (it 
cannot  exceed  ^  f°r  a  valid  solution),  it  is  permissible  to  replace 
us  by  1  +  3z,  and  u2  by  1  +  2i,  which  makes  (47)  linear  in  i 
and  gives  the  relation 

A  —  B  —  2X'  —  2/n 
l=       6/rc  +  4V  -  A 

This  apparently  complicated  analysis  together  with  the  entire 
mathematical  treatment  of  pile  loading  is  given  with  the  idea 
of  affording  a  direct  solution  of  pile  spacing  problems  for  ec- 
centric distributions  of  loading.  The  problems  at  the  end  of 
the  chapter  will  bring  to  bear  the  arithmetic  application  of  the 
literal  equations  just  developed.  The  work  just  shown  of 
determining  the  proper  offset  to  maintain  the  minimum  pile 
spacing  replaces  a  rather  tedious  method  of  trial  and  error.  In 
all  the  above  work  it  is  understood  that  a  uniform  loading  of  the 
several  piles  used  is  the  result  sought. 

For  the  special  case  of  k  =  %,  i.e.  the  resultant  intersects  the 
base  at  the  outer  edge  of  the  middle  third,  and  (45)  becomes 


wJ- 
\n 


(49) 


Table  8  gives  values  of  F  and  H. 

Since  either  method,  theoretically, 
must  give  the  same  density  of  piles,  it 
is  immaterial,  from  the  standpoint  of 
the  number  of  piles  required,  which 
method  is  adopted.  Practically,  how- 
ever, it  seems  simpler  to  use  the  latter 
method  of  distribution  since  the  piles 
are  lined  up  in  both  directions.  In  the 
former,  they  are  in  line  longitudinally, 
only,  i.e.  parallel  to  the  face  of  the  wall 
making  the  work  in  the  field  a  little 
more  cumbersome  than  in  the  latter 
method. 

Occasionally  eccentric  bearing  is 
allowed  on  piles,  the  piles  then  being 


TABLE  8 


i 

F 

H 

.36 

.10 

131.0 

.37 

.14 

65.0 

.38 

.19 

36.8 

.39 

.26 

23.0 

.40 

.33 

15.0 

.41 

.43 

10.0 

.42 

.54 

7.11 

.43 

.69 

5.00 

.44 

.89 

3.51 

.45 

1.20 

2.50 

.46 

1.58 

1.66 

.47 

2.30 

1.10 

.48 

3.67 

.62 

56 


RETAINING  WALLS 


unequally  loaded.-  This  practice  is  far  from  commendable, 
since  a  pile  is,  by  its  very  nature,  a  yielding  support  (unless 
driven  to  absolute  refusal)  and  unequal  settlement  is  unavoid- 
able. Pile  foundations,  and,  in  fact,  all  foundations,  demand 
most  mature  engineering  judgment  in  their  planning  and  con- 
struction and  time  and  money  spent  in  consulting  experienced 
men  on  this  part  of  the  work  is  an  ideal  assurance  towards  a 
safe  and  well-appearing  wall. 

A  problem  at  the  end  of  this  chapter  illustrates  the  application 
of  the  above  analysis  to  a  concrete  case. 

Factor  of  Safety. — It  has  been  seen  that,  as  long  as  the  resultant 
intersects  the  base  inside  the  toe,  there  is  no  danger  that  the  wall 
will  overturn.  Since  the  thrust  is  computed  from  the  maximum 
load  possible  or  anticipated  upon  the  wall,  a  factor  of  safety 
but  little  greater  than  one  seems  ample.  However,  to  insure 
•that  there  will  be  no  tension  in  the  back  of  the  wall,  the  resultant 
should  intersect  within  the  middle  third. 


FIG.  27. — The  retaining  wall  and 
its  foundation. 


FIG.  28. 


The  wall  may  be  divided  into  two  parts;  that  portion  (see 
Fig.  27)  above  the  ground  surface,  retaining  the  fill;  and  the 
foundation  course.  At  the  junction  of  these  two  parts,  that  is,  at 
the  surface  of  the  ground,  the  resultant  should  intersect  at  the 
outer  edge  of  the  middle  third.  This  insures  the  most  economi- 
cal wall  above  the  surface  and  at  the  same  time  prevents  any 
tension  in  the  wall.  The  dimensions  of  the  footing  are  then  solely 
governed  by  the  permissible  soil  pressures. 

The  ratio  between  the  moment  tending  to  resist  the  over- 
turning of  the  wall  and  the  moment  tending  to  overturn  the  wall, 
has  been  termed  the  factor  of  safety  against  overturning. 
Referring  to  Fig.  28  the  overturning  moment  is  Tht  and  the 


DESIGN  OF  GRAVITY  WALLS  57 

resisting    moment  is  Gx  +  Tv[(l  +  fiw  —  Bhtanb].     Denoting 
the  factor  of  safety  by  n 

Gx  +  Tv[(l  +  i)w  -  Bh  tan  6]  =  nTht 

Taking  moments  about  the  point  where  the  resultant  intersects 
the  base  G(x  -  zw)  =  Tht  -  Tv[(l  +  i  -  z)w  -  Bh  tan  b] 
Placing  A  =  Tv[(l  +  i)w  —  Bh  tan  b]  the  two  equations  become 
Gx  +  A  =  n  Th  t;  Gx  -  Gzw  =  Tht+  Tv  zw  -  A.     Combining 
these  two  equations  and  solving  for  n 

_  Tht  +  zw(G  +  Tv)  _         zw(G  +  T,) 

T  Tkt  TnT 

and  conversely 

2  _ 

' 


(6  +  Tv)w 

Prof.  Cain1  advocates  designing  a  wall  for  a  definite  factor  of 
safety  and  recommends  the  following  values  of  n  for  walls  sub- 
jected to  vibratory  loadings,  such  as  walls  adjacent  to  passing 
trains  : 

Walls  less  than  10  feet  high  n  =  3.5 

Walls  from  10  to  20  feet  high  n  =    3 

Walls  around  50  feet  high  n  =  2.5 

Prof.  Hool2  recommends  a  factor  of  safety  of  2  for  the  average 
retaining  wall. 

To  assign  a  definite,  integral  factor  of  safety  against  overturn- 
ing locates  the  position  of  the  resultant  upon  the  base  without 
regard  to  the  character  of  the  distribution  of  the  pressure  upon 
the  soil  that  seems  most  desirable.  Walls  fail  because  of  founda- 
tion weakness  (see  pages  160-163)  rarely  because  the  overturn- 
ing moment  exceeds  the  stability  moment.  An  integral  factor 
of  safety  reverses  this  order  of  importance  and  makes  the  less 
usual  potential  mode  of  failure  the  more  important  criterion.  It  is 
better  procedure  to  decide  upon  the  location  of  the  resultant  of 
the  pressures  and  then  to  learn  what  factor  of  safety  is  to  be  had 
following  the  method  given  on  page  56.  It  is  assumed,  in  figur- 
ing the  factor  of  safety  against  overturning,  that  the  wall  will 
revolve  about  its  toe  as  a  fulcrum.  This  is  possible  only  upon  an 
unyielding  soil;  for  the  other  soils,  as  the  wall  tends  to  turn  on 

1  Trans.  A.  S.  C.  E.,  Vol.  Ixxii. 

2  "Reinforced  Concrete  Construction,"  Vol.  2. 


58 


RETAINING  WALLS 


its  toe,  the  ground  in  the  immediate  vicinity  of  the  toe  will 
crush  so  that  the  conditions  under  which  the  factor  of  safety  was 
computed  will  no  longer  be  valid. 

It  is  doubtful  whether,  in  actual  practice  this  factor  against 
overturning  is  ever  predetermined  or  subsequently  ascertained. 
It  is  well,  however,  as  an  additional  precautionary  measure,  to 
find  its  value  in  the  manner  outlined  before. 

Footing. — The  retaining  wall  proper  may  be  considered  to  end 
at  the  bottom  of  the  fill  retained,  or  at  the  natural  ground 
surface  (see  Fig.  27).  It  is  then  necessary  to  design  a  footing 
that  will  properly  distribute  upon  the  soil  the  pressures  brought 
to  it  from  the  retaining  wall.  If  the  base  of  this  wall  proper  is 
projected  vertically  downwards,  and  if  the 
values  of  Si  and  $2  as  found  on  page  51  in 
equations  39,  40  are  within  the  allowable 
pressures  as  shown  in  Table  7  no  extension 
of  the  base  is  necessary.  When  these  values 
exceed  the  permissible  ones  a  toe  extension 
becomes  necessary.  This  may  be  found  as 
follows :  In  Fig.  29  let  ew  locate  the  position 
of  the  resultant  pressure  and  let  S  be  the 
permissible  soil  pressure.  The  offset  iw  is  that  necessary  to 
make  the  value  of  Si  approach  as  nearly  as  possible  the  allowable 
value  S.  Referring  to  equation  (39),  the  value  of  k  is  now 


FIG.  29.— Toe 
extension. 


k  = 


The  value  of  Si  is 


Place 


(i  +  e)w  _  i  -f  e 
(1  +  i)w  ~  f+1 


2R 


/o       o*  +  * 

TT  I    £i          O  -     j     r 
t)  \  *  +  1 


e\ 


iy  ^1/2.B  =  r 


and  the  above  equation  becomes 

2  -  3e  - 


which  is  a  quadratic  in  i,  which  when  solved  gives 


V12  r(l  -  e)  +  1  -  (2r  +  1) 
2r 


(52) 

(53) 

(54) 

(55) 
(56) 


DESIGN  OF  GRAVITY  WALLS 


59 


The  usual  value,  and  the  one  most  properly  taken  for  e  is 

This  makes  (56)  

~T)  -  (2r  +  1) 


V(S 


2r 


(57) 


which  determines  the  necessary  offset  for  the  base  when  the 
resultant  is  given  in  amount  and  location  and  the  value  of  the 
soil  pressure  intensity  has  been  assigned.  To  aid  in  the  deter- 
mination of  the  offset  when  the  value  of  r  is  given,  Table  9  has 
been  prepared  giving  the  values  of  i  for  a  range  of  values  of  r. 
Some  examples  at  the  end  of  the  chapter  illustrate  the  application 
of  Table  9  to  specific  problems. 

A  less  frequent  requirement,  but  one  which  may  possibly 
exist  (see  problems  at  end  of  chapter)  is  the  determination  of  a 
toe  offset  to  give  a  minimum  intensity  $2  at  the  heel.  With  the 
value  of  k  as  in  equation  (52)  and  from  (40)  after  placing 


=  wSz/2R 

3e  - 


2i 


There  is  obtained  a  value  of  i 

•  =  1  -  s  -  VT-  s(2  -  3e  +  1) 

s 
For  e  =  >£,  this  becomes 


.      1  -  s  -  V(l  -  2s) 


(58) 
(59) 


(60) 


(61) 


Table  10  has  been  prepared  giving  a  range  of  values  of  i  for  the 
possible  variations  in  the  ratio  s. 


TABLE  9 


TABLE  10 


1.00., 

^   .00 

.9   " 

.04 

•ay 

.08 

•7:, 

.12 

•6 

.18 

.5  j 

.24 

.4 

.31 

.375 

.33 

s 

i 

.00 

.00 

.05 

.02 

.10 

.05 

.15 

.09 

.20 

.13 

.25 

.17 

.30 

.22 

.35 

.29 

.375 

.33 

60  RETAINING  WALLS 

The  toe  extension  is  a  cantilever  beam  and  must  be  so  dimen- 
sioned as  to  satisfy  the  shear  and  bending  moment  requirements 
of  such  a  beam.  Let  the  thickness  of  the  toe  be  d.  Since  the 
extension  is  usually  small  in  comparison  with  the  rest  of  the 
footing,  the  distribution  of  soil  pressure  may  be  taken  as  uni- 
formly spread  over  the  toe  and  equal  in  intensity  to  Si,  per  unit 
of  length.  If  fc  is  the  concrete  stress  allowed  in  compression, 
the  external  moment  equated  to  the  resisting  moment  gives 
Sii2w2/2  =  /cd2/6  and  d  =  kiw,  with  k  = 


It  is  necessary  here  to  locate  the  principal  planes  to  determine 
along  what  plane  there  exists  a  maximum  tension,  i.e.,  the  plane 
of  weakness  of  the  step.  The  stresses  on  the  principal  planes 
are  given  by  the  expression  /  =  c/2  ±  \/(c2/4  +  p2).  c  is  the 
unit  compressive  stress  and  p  the  unit  shearing  stress  found  in 
the  body  with  the  axes  corresponding  to  the  axes  of  loading  of 
the  body,  i.e.,  as  in  the  sketch,  vertical  and  horizontal.  In 

slightly  altered  form,  this  may  be  written  /  =  £  ~  ^-J  1  H  —  \  -• 

For  concrete  c  is  large  in  comparison  with  p  and  in  developing 
the  radical  by  the  binomial  theorem  it  will  be  permissible  to  stop 
with  the  second  term,  whence/  =  pz/c,  or  p  =  VC/c)-  The  unit 
shear  is  then  a  geometric  mean1  between  the  tension  and  com- 
pression as  exerted  along  the  vertical  and  horizontal  planes 
of  the  body.  In  the  first  expression  for  the  principal 
stresses,  the  minus  sign  was  taken  since  the  principal  tension 
was  sought. 

The  angle  between  the  principal  tension  plane  and  the 
vertical  plane  is  given  by  tan  -1  (  —  2p/c),  or  using  the  approxi- 

mate relation  between  p  and  c  is  equal  to  tan  ~  12  ^jc'      Upon  the 

recommendation  of  the  special  concre  te  committee  of  the  A.  S.C.E. 
(a  summary  of  which  is  given  later  in  a  section  on  "  Reinforced 
Concrete")  the  ratio  f/c  is  to  be  taken  as  He,  and  this  angle  be- 
comes tan  ~  1  (  —  J-0  or  the  ratio  of  the  extension  to  the  depth  is 
one-half. 

The  maximum  tension  then  exists  along  a  plane  making  a  slope 
of  one  to  two  with  the  vertical.  Again,  it  has  been  demonstrated 
that  the  transmission  of  loading  through  a  solid  is  contained  with- 

1  In  "Reinforced  Concrete"  by  MORSCH,  as  translated  by  E.  P.  GOOD- 
RICH, this  theorem  is  established  by  somewhat  different  an  analysis. 


DESIGN  OF  GRAVITY  WALLS  61 

in  planes  making  an  angle  of  about  30°  with  the  vertical.  For 
both  these  reasons,  good  practice  would  demand  that,  wherever 
possible  the  ratio  of  step  to  depth  for  a  foundation  offset  be  one  to 
two. 

The. maximum  pressure  that  can  be  brought  to  bear  upon 
a  foundation  is  limited  by  the  permissible  bearing  on  the 
masonry,  usually  taken  at  about  thirty  tons  per  square  foot  or 
about  400  pounds  per  square  inch.  From  the  preceding  formula 
for  the  depth  of  step  as  required  because  of  the  bending  moment, 
k  is  then  less  than  2,  so  that  a  step  of  1  to  2  will  always  satisfy 
the  bending  moment  requirements  with  the  above  maximum 
loading.  The  shear  on  the  plane  where  the  toe  joins  the  footing 
is  Siiw/d  =  Si/k.  If  the  shearing  stress  is  taken  as  75  pounds 
per  square  inch,  then  as  long  as  /Si  does  not  exceed  150  pounds 
per  square  inch  or  about  ten  tons  per  square  foot,  a  value  of 
k  =  2,  is  good.  When  the  soil  pressure  does  exceed  this  amount, 
it  will  be  necessary  to  reinforce  the  base. 

For  all  ordinary  soil  pressures,  then,  a  step  of  one  to  two  is 
satisfactory  and  should  be  adopted  for  the  toe  extension. 

A  Direct  Method  of  Designing  the  Wall  Proper. — In  the  ordi- 
nary course  of  design  of  a  gravity  wall,  a  tentative  section, 
governed  by  the  judgment  and  experience  of  the  designer,  is 
selected.  This  is  analyzed  in  accordance  with  the  methods  out- 
lined in  the  preceding  pages.  It  has  been  pointed  out  that  the 
usual  goal  of  the  designer  is  to  select  such  a  section  of  wall  that 
the  resultant  intersects  exactly  at  the  outer  edge  of  the'middle 
third.  As  the  tentative  section  does  not,  at  first  choice,  fulfill 
this  condition,  one  or  more  succeeding  sections  are  chosen  until 
the  final  one  does  meet  this  criterion.  By  using  the  criterion  that 
the  resultant  must  intersect  at  the  outer  edge  of  the  middle  third 
and  by  giving  the  thrust  the  standard  form  of  expression  on  page 
16,  it  is  possible  to  effect  a  direct  solution  of  the  required  dimen- 
sions of  the  wall.  The  analysis  following  develops  an  equation, 
predicated  upon  these  assumptions,  from  which  Table  12  has 
been  prepared.  This  table  covers  the  usual  range  of  the  factors 
controlling  the  wall  section  and  is  to  be  used  in  place  of  the 
method  of  trial  and  error  as  stated  above.  The  numerical  ap- 
plication of  the  table  and  of  the  equations  upon  which  it  is  based 
is  to  be  found  in  the  problems  at  the  end  of  the  chapter. 

The  general  gravity  type  of  wall  is  shown  in  Fig.  30.  The  rec- 
tangular wall,  the  wall  with  a  vertical  front  face  and  the  wall 


62 


RETAINING  WALLS 


with  a  vertical  rear  face  are,  of  course,  but  special  cases  of  this 

general  type. 

In  taking  moments  about  the  outer  edge  of  the  middle  third, 

i.e.,  about  the  point  /,  the  moment  of  the  thrust  must  be  equal 

to  the  wall  moment.  These 
moments  are  found  as  follows : 
Extend  the  sides  of  the  wall 
to  their  intersection  at  A 
project  the  point  A  vertically 
down  upon  the  base,  meeting 
the  base  at  the  point  D.  The 
vertical  distance  that  A  is 
above  the  top  of  the  wall  is  t. 
Let  the  ratio  t/h  be  put  equal 
to  p.  The  front  face  of  the 
wall  makes  an  angle  a  with 
the  vertical;  the  rear  face  (the 
face  adjacent  to  the  earth 
embankment)  an  angle  b. 
Place  tan  a  and  tan  6  equal  to 
M  and  N  respectively.  Tak- 


tip)(M+N)l? 
FIG.  30. — Design  of  gravity  wall. 


ing  moments  about  the  point  D,  the  location  of  the  point  of  ap- 
plication of  the  weight  of  the  wall  with  respect  to  D  is  x,  where 


(62) 


X3        1  +  2p 
The  distance  of  G  from  the  point  0,  i.e.,  from  the  toe  of  the  wall 


is 


(1  +  p)Mh  +  x 
and  from  62  this  becomes 


(63) 


l+3p  +  3p*     \ 
l  +  2p        *) 


This  expression,  locating  the  center  of  gravity  of  a  general  type 
of  gravity  wall  with  respect  to  the  toe  may  be  further  simplified 
by  putting  the  ratio  of  the  upper  to  the  lower  base  equal  to  u. 


Then 


u  =  p/(l  +  p). 


(65) 


DESIGN  OF  GRAVITY  WALLS 


63 


Calling  the  distance  of  the  center  of  gravity  of  the  wall  from  the 
toe,  q,  from  (63)  and  (64) 


(66) 


where 


TJ 
C/2  = 


u 


1  - 


TABLE  11 


u 

C/i       - 

u. 

.0 

2.00 

1.00 

.1 

2.21 

1.12 

.2 

2.46 

1.29 

.3 

2.76 

1.53 

.4 

3.14 

1.86 

.5 

3.67 

2.33 

.6 

4.44 

3.06 

.7 

5.70 

4.30 

.8 

8.22 

6.78 

.9 

15.73 

14.27 

Table  11  has  been  prepared  giving  the  values  of  these  coeffi- 
cients for  the  range  of  values  of  u.  The  table,  and  the  above 
formulas  for  the  center  of  gravity  with  respect  to  the  toe  are 
applicable  to  any  method  of  analyzing  the  wall,  not  only  the 
special  method  now  being  followed. 

The  distance  from  the  outer  third  point  I  to  the  point  of 
application  of  the  force  G  is  x,  where 


x  = 


x-l 


N)h 


(67) 


When  simplified  this  value  becomes 
h  /I  4-  3v  4-  3#2  , 


^^        l  +  2p       ~   '  r+  2pA"  / 

If  the  unit  weight  of  the  masonry  is  m  pounds  per  cubic  foot, 
then  the  value  of  G  is 

a  -  TOfe2a  +  «P)(*  +JV)  (69) 

4 

and  its  moment  about  the  outer  third  point  7  is  Gx,  or 

!}(M+AT)         (70) 


m/i3( ,,,  , 

=  -x-(M(l  +  3p) 


64 


RETAINING  WALLS 


To  determine  the  thrust  moment  resolve  the  thrust  into  its 
horizontal  and  vertical  components  as  shown  on  page  10.  The 
horizontal  component  is  T*,  and  its  value  is 


Th  =  gh2(l  +  2c)/6 

The  vertical  component  is  Tv  and  its  value  is 
Tv  =  gh*(l  +  2c)N/2 


(71) 


(72) 


Taking  moments  about  the  outer  edge  of  the  middle  third  /,  and 
letting  the  thrust  moment  be  M 0. 

Mo  =  TkBh  -  2\[|  (1  +  p)(M      N)h  -  BhN] 

=  ^  (1  +  2c){£  -  N[2(l  +  p)(M  +  N)  -  3BN]}     (73) 

Equating  this  thrust  moment  to  the  stability  moment  of  the 
wall,  putting  the  ratio  of  the  unit  weight  of  the  earth  g  to  the 
unit  weight  of  the  masonry  m  equal  to  s,  and  writing  the  equation 
in  the  form  of  a  quadratic  in  p(M  +  N), 


(74) 


(M  +  AOV  +  Ip(M  +  N)  +  H  =  0 
3M  +  2sN(l  +  2c);H  =  M(M  +  AO 

-  |  [1  -  QMN  -  3N2  4-  3c(l  -  4MN  - 


It  will  be  noticed  that  the  quantity  p(M  -f  N)  is  the  ratio  of 
the  width  of  the  top  of  the  wall  to  the  height  of  the  wall.  Table 
12  has  been  prepared  based  upon  equation  74,  giving  the  ratios 

TABLE  12 


N  =  0.0 

N  =  0.1 

#  =  0.2 

N  =  0.3    |  AT  =  0.4 

N  =  0.5 

M 

c 

c          |         c 

c 

'    \   • 

0 

.2|.4   |0 

.2|.4|0 

.2|.4 

0|.2|.4 

0 

2 

.4|0 

.2 

.4 

.& 

.60 

.70 

.40 

.50 

.58 

.33 

.41 

.47 

.25 

.33 

.37 

.17 

.23 

.28 

.07 

.17 

.23 

0 

.47 

.60 

.70 

.50 

.60 

.68 

.53 

.61 

.67 

.55 

.63 

.69 

.57 

.63 

.68 

.57 

.67 

.73 

.33 

.46 

.56 

.26 

.36 

.44 

.19 

.27 

.34 

.10 

.18 

.24 

.02 

.09 

.14 

.05 

.08 

.1 

.43 

.56 

.66 

.46 

.56 

.64 

.49 

.57 

.64 

.50 

.58 

.64 

.52 

.59 

.64 

.65 

.68 

.22 

.34 

.44 

.15 

.24 

.32 

.07 

.15 

.22 

.06 

.11 

.02 

.02 

.2 

.42 

.54 

.64 

.45 

.54 

.62 

.47 

.55 

.62 

.56 

.61 

.62 

.72 

.13 

.24 

.33 

.05 

.14 

.22 

.05 

.11 

.01 

.3 

.43 

.54 

.63 

.45 

.54 

.62 

.55 

.61 

.61 

.05 

.15 

.23 

.05 

.12 

.01 

.4 

.45 

.55 

.63 

.55 

.62 

.61 

.07 

.15 

.03 

.5 

.57 

.65 

.63 

DESIGN  OF  GRAVITY  WALLS  65 

of  the  top  and  bottom  widths  of  the  wall  to  the  height  of  the  wall  for 
a  sufficient  range  of  values  to  determine  very  closely  the  required 
dimensions  of  any  gravity  type  of  wall,  assuming  that  the  ratio 
of  the  weight  of  the  earth  to  masonry  is  %  (i.e.,  s  =  %)  and 
that  the  resultant  intersects  the  base  of  the  wall  proper  at  the 
outer  edge  of  the  middle  third. 

With  both  M  and  N  zero,  the  wall  is  the  rectangular  type. 
With  M  zero,  the  wall  is  the  vertical  front  and  battered  back  type, 
a  very  popular  type  forming  a  large  percentage  of  all  gravity 
types  built  and  very  efficient  where  maximum  trackage  and 
minimum  easements  are  wanted  (see  page  42).  With  N  zero 
there  is  the  less  usual  type,  but  a  most  economical  one  with 
vertical  back  and  battered  face.  A  slight  face  batter  and  a 
larger  back  batter  make  a  wall  of  economical  section  and  pleasing 
appearance.  It  is  understood  in  selecting  the  dimensions  of  the 
wall  that  a  proper  footing  is  to  be  developed  as  shown  on  the 
preceding  pages,  to  give  the  correct  distribution  of  pressure  upon 
the  foundation. 

The  converse  problem,  given  the  section  of  a  retaining  wall, 
to  locate  the  position  of  the  resultant  pressure  upon  the  base 
may  be  solved  as  follows :  Referring  to  Fig.  30,  with  the  weight 
of  the  wall  G  a  distance  q  from  the  toe  and  the  point  of  applica- 
tion of  the  resultant  pressure  a  distance  zw  from  the  toe  where 
zw  =  01,  as  in  Fig.  23,  take  moments  about  7 

G(q  -  zw)  +  Tv(w  -  BhN  -  zw)  =  ThBh 
and  solving  this  expression  for  z, 

_Gq  +  Tv(w  -  BhN)  -  ThBh  f     . 

(G  +  Tv)w~ 

The  value  of  q  and  of  the  thrust  components  may  be  taken 
from  the  appropriate  equations  and  tables  given  in  the  preceding 
work. 

Revetment  Walls. — The  wall  leaning  toward  the  earth  bank 
which  it  supports,  as  shown  in  Fig.  31,  is  termed  a  revetment  wall. 
It  is  more  of  historic  than  of  present  interest.  Prof.  Cain  has 
shown1  that  when  the  angle  b  is  less  than  10°,  the  ordinary  theory 
of  earth  pressure  as  given  by  the  method  of  the  wedge  of  maxi- 
mum thrust  (see  pages  11-15),  may  safely  be  applied  to  deter- 
mine the  thrust. 

1  "  Earth  Pressure,  Walls  and  Bins,"  pp.  96,  97. 
5 


66 


RETAINING  WALLS 


That  the  wall  be  self-sustaining  while  under  construction,  it 
is  necessary  that  its  center  of  gravity  projected  down,  always 
falls  within  the  base.  To  effect  this,  denote  the  ratio  of  the 
width  of  base  to  height  of  wall  (a  parallelogram  is  the  only  type 
of  section  discussed  in  detail  here)  by  k.  That  the  wall  be 

self-sustaining,  it  is  necessary 
that  k  be  greater  than  tan  b. 
As  in  the  former  pages,  a 
direct  method  of  determining 
upon  the  ratio  k  for  any 
character  of  loading,  predi- 
cated upon  the  resultant  in- 
tersecting at  the  outer  edge 
of  the  middle  third  may  be 
found  for  this  type  of  wall. 
In  the  following  work  the 
earth  pressure  coefficient  is 
K,  defined  by  equation  (25). 
In  view  of  the  fact  that  the 


FIG.  31. — Design  of  revetment  wall. 


angle  b  is  now  negative,  Table  13  has  been  prepared  giving  the 
values  of  this  coefficient  K  for  negative  values  of  the  angle  b. 

The  thrust  moment  is 

T  X  AO  (76) 

From  (24) 

r -,**!+*? 

AO  =  EF  =  ED  -  FD. 
ED  =  Bh  cos(<£'  -  b) 

FD  =  (Bh  tan  6  +  f  kh)  sin  (<£'  -  6) 
o 

and  (76)  becomes 
fi  Kh3(l  +  2c)[3J5  cos  (0'  -  b)  -  (SB  tan  b  +  2fc)  sin(0;  -  6)] 

The  stability  moment  of  the  wall  (both  of  the  moments  are 
taken  about  the  outer  edge  of  the  middle  third,  i.e.  0)  is 


mkh* 


\  kh* 

tan  b  +  kh/2  -  kh/Z)  =  m  jr  (3  tan  b  + 


k) 


Equating  these  two  moments,  and  writing  the  resulting  equa- 
tion as  a  quadratic  in  k 


+  Rk  =  S 


(77) 


DESIGN  OF  GRAVITY  WALLS 


67 


where 


R  =  3  tan  b  +  2s(l  +  2c)  sin  (<*>'  -  6) 

i*  **•*•*>  2? 


(78) 
(79) 


s  is  the  ratio 


m 


TABLE  13 


TABLE  14 


b 

4>'  =  0° 

*'=15° 

<£'  =  30° 

0° 

.33 

.30 

.29 

5° 

.30 

.27 

.26 

10° 

.27 

.24 

.23 

<£'  =  0        |      </=15° 

0'  =  30° 

6  =  5° 

6=10° 

i  =  5°|6  =  10°  |b  =  5° 

6=10° 

0 

.35 

.25 

.30 

.21 

.23 

.17 

.2 

.47 

.37 

.40 

.29 

.31 

.23 

.4 

.57 

.46 

.48 

.37 

.37 

.29 

Table  14  gives  a  series  of  values  of  the  ratio,  fc,  based  on  the 
above  equation  for  several  values  of  b  and  <£'.  Revetment  walls 
are  usually  built  of  stone  masonry,  presenting  quite  a  rough 
surface  adjacent  to  the  earth  bank,  and  it  is  therefore  safe  to 
allow  the  usual  value  of  <j>'  (about  30°).  Revetment  walls, 
because  of  construction  difficulties  are  rarely  built  of  concrete. 
If  concrete  should  be  used,  its  smooth  surface,  together  with  the 
possibility  of  lubrication  due  to  water,  makes  it  inexpedient  to 
allow  for  any  frictional  resistance  between  the  wall  and  the  adja- 
cent earth. 

Problems :  Gravity  Walls  and  Foundations 

NOTE. — A  comparative  study  of  various  sections  of  walls,  with  illustra- 
tive plates,  is  given  in  a  pamphlet  published  by  the  Engineering  News,  1913, 
entitled  "Comparative  Sections  of  Thirty  Retaining  Walls  and  Some  Notes 
on  Design,"  by  E.  H.  CARTER. 

1.  A  wall  with  a  slight  face  batter  and  battered  back,  25  feet  high,  sup- 
ports a  fill  level  with  its  top  and  subject  to  a  uniformly  distributed  load  of 
600  pounds  per  square  foot.  What  is  the  necessary  width  of  the  base  as- 
suming that  the  top  width  is  taken  as  2'  6"  wide?  Determine  the  offset  of 
its  footing  that  the  toe  pressure  shall  not  exceed  6000  pounds  per  square  foot. 
What  is  the  factor  of  safety  of  the  wall?  If  the  method  of  the  maximum 
wedge  of  sliding  is  used  where  is  the  point  of  application  of  the  resultant 
located  and  what  is  the  factor  of  safety  (a)  when  the  angle  of  friction  is 
assumed  as  30°  (&)  and  when  it  is  assumed  as  0°  between  earth  and  back 
of  wall? 

The  equivalent  surcharge  to  a  load  of  600  pounds  per  square  foot  is  six 
feet,  whence  the  value  of  c  is  %5,  or  0.24.  The  ratio  of  top  width  to  height 
is  2.5/25  or  0.10.  By  interpolation  in  Table  12  the  values  M  =  0.067  and 


68  RETAINING  WALLS 

N  —  0.5  satisfy  the  given  arguments  and  the  resulting  width  of  base  is 
A  (0.1  +  0.5  +  0.067)  =16.7  feet.  The  face  batter  is  %"  to  the  foot  and 
the  rear  6"  to  the  toot. 

To  obtain  the  proper  soil  distribution,  the  weight  of  the  wall  (taking  the 
masonry  unit  weight  150  pounds  per  cubic  foot)  is  35.9  kips  (i.e.  a  kip  is  a 
1000  pound  unit).  The  vertical  component  of  the  thrust  is  (Eq.  72)  Tv  = 
23.1.  The  vertical  component  of  the  resultant  pressure  upon  the  base  is 
the  sum  of  these  two  forces  or  is  equal  to  35.9  +  23.1  =  59.0  kips.  From 
(54)  r  =  0.85  and  from  Table  9,  i  =  0.057,  whence  the  necessary  projection 
is  iw  or  1'  0".  Since  the  wall  foundations  are  carried  down  about  four 
feet  to  prevent  fiost  action  and  surface  water  erosion,  the  step  of  one  foot 
to  four  feet  is  a  satisfactory  one. 

Fiom  (50)  referring  to  Fig.  28,  B  from  Table  3,  is  0.39  whence  t  =  0.39  X 
25  +  4.0  =  13.75.  zw  =  >£  of  16.7  +  iw  =  6.56  and  the  horizontal  com- 
ponent of  the  thrust  from  (71)  is  15.4  whence  the  factor  of  safety  =  1  + 
1.8  =  2.8,  a  satisfactory  one  from  Prof.  Cain's  recommendations,  page  57, 
but  clearly  without  significance,  unless  taken  in  conjunction  with  the  loca- 
tion of  the  resultant  and  with  the  manner  of  the  distribution  of  pressure 
upon  the  soil. 

By  the  sliding  wedge  method  the  horizontal  component  of  the  thrust  is  T  cos 
(6  +  <£'),  with  T  as  given  in  (24).  For  N  =  0.5,  6  =  26°  34'  and  from 
(25)  K  =  0.60.  Th  and  Tv  are  then  15.4  and  23.3  respectively.  (Cf.  cor- 
responding values  by  other  method.)  The  location  of  the  weight  of  the 
wall  G  is  obtained  from  (66)  and  Table  11  with  u  =  0.10/0.567  =  0.18. 

25 
q  =  -y  (2.41  X  0.067  +  1.25  X  0.5)  =  6.55,  whence  from  (75)  z  =  0.364, 

not  at  large  variance  with  the  value  of  i  +  e  in  the  Rankine's  method. 
The  toe  pressure  is  from  (53)  6.4  kips,  approximating  with  sufficient  exact- 
ness the  result  obtained  in  the  suggested  standard  method  of  obtaining 
the  thrust. 

If  the  frictional  resistance  between  earth  and  masonry  is  ignored,  K  = 
0.64  and  Th,  Tv  are  respectively  26.5  and  13.2.  With  the  revised  values, 
z  =  0.163,  a  very  unsatisfactory  result.  If  the  section  of  wall  is  changed 
to  give  a  value  of  z  =  0.333  by  the  last  method,  a  much  heavier  section  of 
wall  results,  showing  the  costly  effect  of  omitting  the  consideration  of  fric- 
tional action  of  the  earth  upon  the  back  of  wall.  All  the  standard  sections 
exhibited  in  the  above-mentioned  pamphlet  would  develop  high  tension  at 
the  heel  of  the  wall  and  a  high  bearing  at  the  toe  leading  to  the  disfiguration, 
if  not  destruction  of  the  wall  were  they  designed  in  accordance  with  the 
maximum  wedge  of  sliding,  ignoring  frictional  action  between  the  earth 
and  wall.  The  sections  are  all  extensively  used  in  actual  practice  with 
excellent  results. 

Allowing  for  frictional  resistance  between  earth  and  wall  the  factor  of 
safety  is  3;  ignoring  such  action  the  factor  becomes  1.5,  i.e.,  such  favorable 
consideration  doubles  the  factor  of  safety. 

2.  A  standard  wall  for  highways,  is  to  be  built,  with  a  face  batter  of  1>£" 
to  the  foot  and  a  back  batter  of  4"  to  the  foot.  Give  a  section  with  the 
proper  tabular  dimensions.  Also  prepare  plans  for  the  proper  foundation 
dimensions  for  (a)  coarse  sand  and  clay,  well  compacted,  permissible  bearing 


DESIGN  OF  GRAVITY  WALLS 


69 


4  tons  per  square  foot,  (6)  coarse  sand,  permissible  bearing  3  tons  per  square 
foot  (c)  fine  sand,  where  a  maximum  intensity  of  toe  pressure  is  2  ton  per 
square  foot  and  a  minimum  intensity  of  heel  pressuie  is  0.5  tons  per  square 
foot.  Also  give  a  pile  foundation  section,  allowing  twenty  tons  per  pile. 

With  the  batters  as  given,  M  -  0.125,  and  N  =  0.333.  For  highways, 
an  average  uniformly  distributed  load  of  500  pounds  per  square  foot  will 
safely  provide  for  the  heavy  surface  loadings.  Then  for  h  =  15,  c  =  0.33; 
for  h  =  20,  c  =  0.25;  for  h  =  25,  c  =  0.20;  lor  h  =  30,  c  =  0.17.  From 
data  obtained  from  Table  12,  the  following  table  of  top  and  bottom  widths 
of  wall  has  been  prepared,  (d  is  the  top  width,  b  the  base  width.) 


h 
15 
20 
25 
30 


d 

2'  5" 
2'  8" 
3'0' 
3' 4" 


b 

9'  3" 
11'  10" 
14'  5" 
17'  0" 


Following  the  preparation  of  this  table,  a  similar  one  may  be  prepared,  giving 
the  data  necessary  to  compile  the  required  toe  extensions  for  the  several 
allowable  pressuies. 


Si  =  4  tons 

Si  =  3  tons        ||  Si  =  2  tons 

Sz  =  0.5  ton 

h 

fi 

m 

T 

r> 

1  1 

r\i\iw\     r 

i 

ft,     1|     r     \     i 

s 

* 

iw 

15 

13.1 

6.2 

6.2 

19.3 

'  * 

* 

* 

.96 

.02 

.24 

.17° 

l'-6" 

20 

21.7 

10.0  10.0 

31.7 

* 

* 

* 

.75 

.10 

.19 

.13° 

l'-6" 

25 

32.7 

14.6  14.6 

47.3 

* 

.90 

.035 

0'-6" 

.61 

.18° 

.15 

.09 

2'-6" 

30 

45.7 

20.1  20.1 

65.8 

* 

.78 

.08 

l'-4" 

.52 

.23° 

.13 

.08 

4'-0" 

I 

i 

*No  offset  necessary.     °This  value  governs. 

For  the  coarse  sand  and  clay  bottom  (4  tons  per  square  foot)  no  toe  extension 
is  necessary. 

In  preparing  typical  pile  foundation  plans,  it  is  assumed  that  the  piles 
will  be  in  line  both  transversely  and  longitudinally  (Case  II). 
h  =  15'.  Assume  two  piles  to  a  section.  If  the  rows  are  m  feet  apart, 
and  with  a  beaiing  value  of  40  kips  each,  the  necessary  spacing  of  the  rows 
is  SQ/R  =  4.15;  therefore  space  these  rows  on  four  foot  centers.  The  total 
load  on  each  row  is  then  4/2=4  XJL9.3  =  77 K.  With  a  value  of  k  =  }$, 
Eq.  49  is  applicable  and  Zi  =  wv/0.5  =  6.55.  The  location  of  the  pile  is 
at  the  center  of  gravity  of  this  triangle  or  at  a  distance  %j  X  6.55  from  the 
heel.  The  pile  is,  accordingly  4'  4"  from  the  heel.  The  other  pile  is  at 
the  center  of  gravity  of  the  trapezoid  bounded  by  the  toe  and  the  line  li. 
The  center  of  gravity  of  the  trapezoid  may  be  found  in  a  manner  similar  to 
the  location  of  the  center  of  gravity  of  the  earth  pressure  triangle  Fig.  5.  The 
6.55 


value  of  c  is  here 


— TT^  =  2.4  and  the  value  of  B  from  Table  3  is  0.47. 


9.25  -  6.55 

The  center  of  gravity  is  then  4Koo  of  the  distance  2.70  from  the  toe,  or 
approximately  1'  3"  from  the  toe.  It  is  safe,  generally  to  take  the  pile  at 
the  center  of  the  trapezoid,  the  error  being  one  of  a  few  inshes  only. 


70 


RETAINING  WALLS 


h  =  20'.  Assuming  two  piles  in  a  row  here,  with  the  value  of  R  =  31 '.7 
gives  a  spacing  between  rows  of  2.5'  which  is  too  close  to  space  the  piles; 
therefore  three  piles  are  taken.  With  this  value  m  =  3.8  and  may  be  taken 
as  4'.  To  ascertain  whether  a  toe  extension  is  necessary  to  permit  a  mini- 
mum spacing  of  3'  between  the  piles  adjacent  to  the  toe,  the  value  of  i  from 
equation  (48)  with  X'  =  6/11.83  =  0.506;  e  =  M;  and  n  =  3,  is  found  to 
be  0.073.  The  required  toe  extension  is  thus  0.073  X  11.83  =  0.86  or  10". 

i  4-  e 
The  corresponding  value  of  k  is .  =  0.37.     From  Table  8,  F  and  H  are 

respectively  0.14  and  65.0.     Applying  equation  (45) 

11  =  (11.83  +  0.83)0. 14(VTT22  -  1)  =  6.75 

12  =  12.66  X  0.14(Vl  +44-1)  =  10.2 

The  pile  adjacent  to  the  heel  is  4'  0"  from  the  heel,  and  bearing  in  mind 
the  remarks  previously  made,  the  other  two  piles  are  8'  6"  from  heel  and 
I'  2"  from  toe  respectively. 

h  =  25'.  Here  R  =  47.3.  With  an  assumed  number  of  piles,  4  to  a 
row,  the  required  spacing  between  rows  is  found  to  be  3'  6".  To  get  the 
toe  extension,  X'  =  6/14.42  =  0.414.  Accordingly  i  =0.16  and  the  toe 
extension  is  0.16  X  14.42  =  2'  4".  For  simplicity  make  this  2'  0".  kis 

0.14  +  0.33 
then — — =  0.41  and  F  and  H  are  0.43  and  10.00  respectively. 

From  (45) 


li  =  (14.42  +  2.0)  X  0.43  X  0.87 
1,  =  16.42  X  0.43  X  1.45  =  10.2 
Is  =  16.42  X  0.43  X  1.92  =  13.5 


6.11 


Coarse  Sand  and  Clay 

Coarse  Sand 

Fine  Sand  — 

Piles  - A4-1 

When  A  is  overtn.  Step  Base    f 
as  shown.  Li  J 

Reinforce  Base  when  A  is         L-|— 
overZ-fr.  k 


FIG.  32. 


h 

d 

b 

A» 

A* 

A3 

A4 

15 

2-5" 

9-3" 

0 

0 

1-6" 

0 

20 

2-8" 

ll-'ltf 

0 

0 

1-6' 

•1-0" 
2-0" 

25 

3-0" 

.14-5" 

0 

0-6" 

2-6" 

30 

£4" 

17-0" 

0 

1-4" 

4-0"' 

3-0 

The  pile  adjacent  to  the  heel  is  4'  0"  from  the  heel;  the  next  is  8'  0*;  the 
third  12  0"  and  the  face  pile  is  )'  6"  from  the  toe,  this  spacing  closely  ap- 
proximating the  centers  of  the  several  pressure  trapezoids. 

h  =  30'.  R  =  65.8.  With  5  piles  in  a  row,  the  required  spacing  between 
rows  is  found  to  be  3'.  e  is  0.353  and  i  =  0.2.  The  toe  extension  may  be 
taken  as  3'  0".  The  value  of  k  is  0.42  and  F  and  H  are  0.54  and  7. 1 1  respec- 
tively. Then  1:  =  6.0,  12  =  10.4;  13  =  13.9;  14  =  17.2.  The  piles  are 
spaced  3'  0";  8'  0";  12'  0";  and  15'  6"  from  the  heel  and  the  face  pile  1'  6" 
from  the  toe. 


DESIGN  OF  GRAVITY  WALLS 


71 


Figs.  32  and  33  show  the  wall  proper  and  its  foundations.  It  is  under- 
stood, of  course,  that  in  preparing  actual  plans  for  construction  that  the 
plans  will  cover  a  much  closer  variation  in  the  heights. 

3.  A  wall  of  " quaker"  section,  25  feet  high  is  to  rest  upon  a  rock  bottom. 
A  surcharge  of  500  pounds  per  square  foot  extends  to  the  back  of  the  wall. 
It  will  be  permissible  to  let  the  resultant  intersect  at  the  outer  %  point. 
Any  tension  developed  in  the  wall  because  of  this  location  of  the  resultant 
must  be  carried  by  steel  reinforcement. 


rEiJ 


h-J5' 


iliT- 


-,  -j-*--f  H 

i-  d i 


h=20' 


^T7 


FIG.  33.— Pile  layout. 

In  order  to  effect  a  direct  design  of  a  wall  of  this  section,  when  the  position 
of  this  resultant  is  at  the  outer  quarter  point,  it  will  be  necessary  to  proceed 
as  in  the  present  chapter.  Referring  to  Fig.  30  and  equation  (62)  with 
M  =  0, 

_h     1  +  3p  +  3p«  „ 
*  ~  3  1  +2»       ^ 


From  the  quarter-point  to  D  is 


(i  + 


and  G  from  (69)  is 


4 

-f  2p)N 


The  lever  arm  of  G  about  the  quarter-point  is  then 
h  1  +  3p  +  3p2  Ar       hN  ,,  hN 

3         l+2p    -  *  -  '  -T  (1  +  2p;  =12  - 

And  the  moment  of  G  about  this  point  becomes 

m  —n-f-  (1  +  3»  +  6»2) 


+2p 


The  horizontal  and  vertical  components  of  the  thrust  are  respectively  from 
(71,  72) 

ghz(l  +  2c) 
6 


72 


RETAINING  WALLS 


The  lever  arm  of  the  horizontal  component  is  simply  Bh  and  that  of  the 
vertical  component  is 


-  BhN  =        [3(1 
The  overturning  moment  due  to  the  thrust  is 
?^l.±2£)  Bh  _  t 


-  4B] 


Equating  the  stability  and  overturning  moments 

mN*(l  +  3p  +  6p2)  =  0(1  +  2c){4£  - 
and  replacing,  as  before  g/m  by  s 

+  3c)  -  9s(l 


or 


4s(l 


where 


-I-  ZpNI  +  J=Q 
7  =  Ml  +  3s  (1  +  2c)] 
J  =  N*[l  +  s(5  +  6c)]  -  4s(l  +  3c)/3 
Solving  the  quadratic 

24/  -  37} 


TABLE  15 


.     JV 
c    ^^^ 

0 

0.1 

0.2 

0.3                       0.4 

.31 

.23 

.16 

.08 

0 

.39 

.41 

.43 

.46 

.48 

.38 

.30 

.21 

.11 

.2 

.49 

.48 

.50 

.51                  .51 

.46 

.35 

.25                  .14 

.4 

.57 

.56 

.55 

.55                  .54 

To  establish  a  table  (see  Table  15)  take  the  ratio  s  at  its  usual  value  %j. 

To  apply  the  results  of  the  above  to  the  problem  at  hand  note  that  c  = 
5/25  =  0.2.  Let  the  coping  width  be  placed  at  2  feet.  From  the  variation 
of  the  top  and  base  ratios  as  seen  in  the  table  the  base  width  may  be  taken 
as  0.5  X  25  or  12.5  feet. 

To  determine  the  character  of  the  stresses  in  the  wall  it  becomes  neces- 
sary to  locate  the  line  of  resultant  pressures,  or  thrusts  in  the  wall.  This 
is  best  done  graphically.  The  wall  is  divided  up  into  sections  five  feet  high. 
The  weight  and  thrust  upon  each  section  is  determined  as  shown  in  Fig.  34. 
The  points  of  application  of  each  of  these  forces  are  found  as  follows:  the 
center  of  gravity  of  the  masonry  trapezoids  is  taken  from  equation  (66) 
and  table  11,  where  q  =  hUzN/3,  or,  since  N  =  0.42  and  h  is  constant  and 
equal  to  5  for  each  section, 

q  =  0.7  U* 


DESIGN  OF  GRAVITY  WALLS 


73 


For  the  five  sections  starting  from  the  top  the  ratios  of  the  upper  to  lower 
base  (u)  are  respectively  0.50;  0.66;  0.74;  0.79  and  0.83  and  the  correspond- 
ing values  of  q  are  then  1.64;  2.60;  3.60;  4.45  and  5.7.  The  weights  of  these 
sections  are  respectively  2.3;  3.8;  5.4;  7.0  and  8.6.  The  centers  of  gravity 
of  the  thrust  triangles  are  found  most  easily  from  table  3,  using  the  proper 
value  of  c.  Since  the  surcharge  is  5  feet,  the  respective  values  of  c  to  be 
used  in  determining  the  values  of  B  to  locate  the  point  of  application  of  the 
thrusts  are  1;  2;  3;  4  and  5  and  the  point  of  application  above  the  base  of 


FIG.  34. 

each  trapezoid  is  2.2;  2.35;  2.38;  2.41  and  2.42.     For  the  sake  of  simplicity 

and  to  reduce  the  number  of  lines  to  be  drawn  the  resultant  of  each  of  these 

two  forces  will  be  used.     To  determine  the  line  of  thrusts  it  is  most  easy  to 

apply  the  principles  of  the  funicular  polygon.     The  load  polygon,  at  the 

right  of  the  figure  is  first  drawn.     The  direction  of  each  of  the  resultants 

is  found  to  be  the  same  and  parallel  to  the  total  resultant  at  the  base  of  the 

wall.     The  pole  of  the  polygon  is  taken  at  convenience  and  the  r&ys  are 

drawn  to   the  individual  resultants.     The 

funicular  polygon   is   drawn  in   the  usual 

manner  and  the  location  of  the  resultant 

thrust  upon  each  section  is  determined  by 

the  intersection  of  the  corresponding  ray 

with  ray  1,  extended  when  necessary.     A 

line  through  this  intersection   parallel   to 

the  direction  of  the  resultant  shown  in  the 

load  polygon  determines  this  location  of  the 

resultant  thnst.     The  vertical  components 

of  the  resultant  pressure  upon  the  base  of  each  section  is  scaled  from  the 

load  polygon. 

Whenever  the  point  of  application  of  the  resultant  thrust  lies  within  the 
outer  third  there  is  tension  developed  at  the  rear  of  the  wall  and  it  is  neces- 
sary to  determine  this  amount  and  supply  sufficient  steel  rod  reinforcement 
to  take  care  of  this  tension,  it  being  assumed  that  the  wall  shall  take  no 
tension  whatsoever.  From  an  inspection  of  the  figure  it  is  seen  that  above 


FIG.  35. — Amount  of  tension 
in  wall. 


74 


RETAINING  WALLS 


the  line  a  the  resultant  pressure  lies  within  the  middle  third  and  there  is 
consequently  no  tension  in  the  concrete  above  this  point. 

From  Fig.  35,  the  steel  area  necessary  to  take  the  tensile  stresses  developed 
is  that  required  by  the  shaded  portion.  The  area  of  this  portion  is  x  S*/2. 
From  (41), 

w  1  -  3k 


X    =   7T 


and  S2  from  (40)  is 

S2 
The  area  is  then 

where 


2R 

w 


3    1  -  2k 


(1  —  3fc),  disregarding  the  negative  sign. 


R  (1  - 


3      I  -2k 

1   (1  -  3fc)2 
"31-  2k 


Table  16  gives  a  list  of  the  values  of  V  for  several  values  of  fcless  than 

TABLE  16 


.33 

.00 

.30 

.01 

.25 

.04 

.20 

.09 

.15 

.14 

.10 

.20 

.05 

.27 

FIG.  36. 


The  values  of  R  as  determined  from  the  load  polygon  for  each  of  the  sec- 
tions 6,  c,  d  and  e  are,  respectively  10.3;  19.2;  31  and  45.5.  The  correspond- 
ing values  of  k  (by  scaling)  are  %oJ  1%iJ  l%2  and  1^0-  (Note  that 
this  last  value  of  k  affords  a  check  upon  the  algebraic  method  of  obtaining 
the  dimensions  of  the  wall;  having  assumed  the  location  of  the  resultant  at 
the  outer  quarter  point)  or  0.26;  0.25;  0.23  and  0.25  for  which  the  values 
of  V  are  0.04,  0.04;  0.06  and  0.04.  The  total  area  of  the  sections,  or  rather, 
the  total  tension  that  must  be  taken  by  the  steel  are  respectively  410;  770; 
1860  and  1820.  Assuming  that  the  steel  rods  can  take  16,000  pounds  per 
square  inch,  a  %  inch  square  rod  every  12''  will  afford  sufficient  section  to 
take  the  maximum  stress.  Since  it  is  not  necessary  to  have  this  amount 
of  metal  to  the  plane  b,  the  rods  will  be  spaced  at  12"  centers  to  the  plane  c 
and  at  24"  centers  to  the  plane  a.  The  rods  will  be  placed  3"  from  the  back 
of  the  wall.  Figure  36  shows  the  final  wall  section. 

4.  A  dry  rubble  wall,  35  feet  high  with  front  face  battered  one-inch  to 
the  foot  and  rear  face  battered  4><j  inches  to  the  foot  weighs  125  pounds 
per  cubic  foot.  The  earth  surface  is  horizontal  and  is  subject  to  a  live  load 
of  500  pounds  per  square  foot.  The  soil  pressure  must^not  exceed  3  tons 
per  square  foot.  Determine  the  proper  wall  and  footing  dimensions. 


DESIGN  OF  GRAVITY  WALLS  75 

For   this   problem    Af  =  K2   and    N  =  %.     Referring  to   (74),   s  = 
10%25   =  0.8  and  c  =  %5  =  M- 
/  =  M2  +  2  X  0.8(1  +  WH  =  1.02 


1  /I       3\       0.8  L        ft      1    V3  32 

T1  XT2X8~        X.8« 


The  quadratic  now  becomes,  after  putting  p(M  +  N)  =  x 

x*  +  1.02z  -  0.15  =0 
From  which  x  =0.13 

The  base  ratio  is  0.13  +  (Af  +  N)  =  0.59.  The  top  and  base  width  of 
the  wall  are  then  0.13  X  35  =  4  feet  6  inches  and  0.59  X  35  =  20  feet. 

Note  that  for  a  wall  of  concrete  or  rubble  masonry  weighing  150  pounds 
per  cubic  foot  the  top  and  base  ratios  for  the  same  conditions  as  the  wall 
in  the  problem  are,  from  table  12,  0.07  and  0.53  or  the  widths  become  2'6" 
and  18'6"  respectively.  The  area  of  this  latter  wall  is  85  per  cent,  of  the 
area  required  of  the  dry  rubble  wall.  That  is,  15  per  cent,  more  area  is 
required  when  the  unit  weight  of  the  masonry  is  decreased  15  per  cent.  —  a 
result  quite  obviously  expected. 

The  vertical  component  of  the  thrust  is  from  (72) 


and  in  the  variables  of  this  problem 

I'OO  X  1.286  X  352  X  0.375/2  =  29.5 
The  weight  of  the  wall  is 

35  X  125  X  4'5  +  2°  =53.5kips. 

The  total  vertical  component  of  the  resultant  pressure  upon  the  base  is 
83  kips.  The  permissible  soil  pressure  intensity  is  6  kips  per  square  foot. 
From  (54)  r  -  wSi/2R  =  0.723  and  from  Table 
9  with  this  value  of  r,  the  necessary  value  of  i 
is  0.11.  The  toe  extension  is  0.11  X  20  =  2'3". 
As  indicated  on  page  61  the  depth  of  footing 
will  be  4'6".  The  complete  section  of  the  wall 
is  shown  in  Fig.  37.  In  conformity  with  the 
usual  practice  the  coping  is  made  of  concrete 
and  carried  back  2'6".  . 

6.  A  rectangular  wall  is  to  line  a  rock  cut 


twenty  feet  high   and  may  be  subjected  to 

hydrostatic  pressure  up  to  one-half  of  the  full      2L2>"—-'    ~"?0 


water  pressure.      Determine  the  necessary  wall     FIG.  37. — Dry  rubble  wall, 
thickness.     To  avoid  the  necessity  of  placing 

steel  in  the  wall  the  point  of  application  of  the  resultant  should  lie  at  the 
outer  third  point. 

One-half  fluid  pressure  is  31  pounds  per  cubic  foot.     For  a  wall  with 
vertical  back  the  lateral  earth  pressure  has  an  intensity  of  H  of  the  vertical 


76  RETAINING  WALLS 

and  with  earth  at  100  pounds  per  cubic  foot  (the  usual  value)  this  intensity 
is  33  pounds  per  cubic  foot.  The  problem  is  then  merely  to  find  a  wall 
satisfying  an  earth  pressure  thrust  as  given  in  (14)  with  c  =  0  and  K  =  %. 
From  Table  12  with  N  =  M  =  0  and  c  =  0,  the  required  ratio  of  base  to 
height  is  0.47.  The  necessary  thickness  of  the  wpll  for  the  conditions  of 
the  problem  is  9'6". 

6.  A  wall,  whose  resultant  brings  a  vertical  component  of  35  kips  per 
linear  foot  of  wall  located  at  the  outer  third  point  must  have  a  uniform  dis- 
tribution of  loading.     The  base  of  the  wall  proper  is  12  feet  wide.     Design 
the  foundation. 

For  a  uniform  distribution  the  resultant  must  be  at  the  center  of  the  foot- 
ing. Since,  under  the  conditions  of  the  problem  the  location  of  the  point 
of  application  of  the  resultant  is  8  feet  away  from  the 
heel,  the  footing  must  be  16  feet  wide,  necessitating 
a  four-foot  toe  extension.  The  uniform  intensity  of 
pressure  is  then  3^fe  °r  2.2  kips  per  linear  foot. 
The  shear  at  the  cantilever  support,  see  Fig.  38,  is 
for  uniform"^  dltrb8-  4  X  2.2  =  8.8  kips.  Since  the  usual  depth  of  footing 
uted  base  load.  is  f°ur  fee*?  to  bring  the  base  of  the  wall  below  the 

fiost  line,  the  step  will  be  made  4  feet  high  as  shown 

in  figure.  The  unit  shear  is  8800/(48  X  12)  =  15  pounds  per  square  inch. 
The  cantilever  moment  at  the  same  point  is  8800  X  24  =  211,000  inch 
pounds.  The  section  modulus  is  bdz/Q  =  8600.  The  tension  at  the 
lower  edge  of  the  base  is  then  211,000/8600  or  24  pounds  per  square 
inch.  Clearly  no  reinforcement  is  necessary.  For  the  economy  of  the 
material  the  step  will  be  made  in  two  sections  oi  like  dimensions.  The 
shear  is  now  4400/(24  X  12)  =15  pounds  per  square  inch  and  the  moment 
is  220  X  24  =  53,000.  The  section  modulus  is  12  X  242/6  =  1150.  The 
unit  tension  is  53,000/1150  =  46  pounds  per  square  inch.  The  safe  value 
is  slightly  less  than  this  (40  pounds  per  square  inch)  but  this  variation 
from  the  safe  stress  is  a  permissible  one  and  no  reinforcement  will  be  added. 

7.  In  the  wall  of  problem  3  an  opening  is  to  be  placed  as  shown  in  Fig.  39. 

Determine  whether  it  is  necessary  to  rein-  _ 

force  the  section  of  the  wall  to  make  it 

span  safely  the  opening. 

The  resultant  load  per  lineal  foot  of  the 
wall  was  found  to  be  45.5  kips  per  foot. 
The  span  in  the  clear  is  20  feet. 


The  wall  is  on  a  rock  footing,  so  that  set-     /?0c/r  %  V      to' 
tlement  is  improbable  and  it  seems  reason- 
ably safe  to  take  the  wall  as  a  fixed  beam,  FlG   39. 
with  moment  wl2/l2  at  the  support.    How- 
ever, since  the  wall  may  crack  near  the  supports  for  some  reason  unfore- 
seen, it  is  better  to  investigate  the  stresses  at  the  center  of  the  span  on 
the  assumption  that  the  beam  is  a  simple  one,  and  to  make  provision  for 
stresses  at  the  support  in  accordance  with  the  assumption  of  a  fixed 
beam.    As  a  simple  beam  the  moment  is  45.5  X  400/8  =  2275  kip  feet. 
As  a  fixed  beam  the  moment  is  45.5  X  400/12   =  1520  kip  feet.      The 
total  moment  is  then  45.5  X  400/12  =  1520  kip  feet.     The  shear  is  455 


DESIGN  OF  GRAVITY  WALLS 


77 


kips.     The  area  of  the  wall  is  26,100  square  inches  giving  a  unit  shear  o 
455,000/26,100  =  17  pounds  per  square  inch. 

The  apex  of  the  section  (produced)  is  about  5  feet  above  the  top  of  the 
wall.  Analogous  to  the  location  of  the  center  of  gravity  of  the  thrust 
triangle  the  center  of  gravity  of  the  beam  section  is  located  a  distance  Bh 
above  the  base,  where  with  c  =  0.2,  B  =0.38  from  Table  3  and  the  center 
of  gravity  of  the  section  is  located  0.38  X  25  or  9.5  feet  above  the  base. 

From  the  "Carnegie  Handbook"  (p.  137)  the  moment  of  inertia  of  the 
section  about  its  center  of  gravity  axis  is  given  by  an  expression 


I  = 


36(6  +  60 


where  d  is  the  depth  corresponding  to  h  here  and  6  and  61  are  respectively 
the  lower  and  upper  bases. 

Using  foot4  units,  the  moment  of  inertia  of  the  given  section  is 

253(22  +  4  X  12.5  X  2  +  12.52) 


I  = 


36(2  +  12.5) 


^  =  7800. 


To  the  extreme  fibre  in  tension  at  the  center  of  the  span,  the  distance 
is  9.5  feet,  and  the  section  modulus  becomes  7800/9.5  or  820. 

The  unit  tension  per  square  foot  is  then 
2,275,000/820  =  2780  or  19  pounds  per 
square  inch.  No  reinforcement  is  then 
necessary.  Over  the  supports  the  maximum 
tension  occurs  at  the  top  of  the  wall.  The 
distance  of  the  extreme  fibre  is  now  25  — 
9.5  —  15.5  and  the  coiresponding  section 
modulus  is  7800/15.6  =  500.  The  unit 
tension  per  square  foot  is  1,520,000/500  = 
3040  and  the  unit  tension  per  square  inch 
is  21  pounds. 

While  no  steel  is  necessary  theoretically,  a 


Original 
Wall--''' 


FIG.  40. — Reconstruction  of 
gravity  wall. 


prudent  engineer  may  specify  light  reinforcement  over  the  supports,  at  the 
top  of  the  wall  and  along  the  bottom  of  the  wall  from  support  to  support 
(see  Fig.  39;. 

Some  Examples  in  Recent  Practice 

1.  Wall,  Reinforced  on  Bottom  on  Account  of  Threatened  Settlement, 
Engineering  Record,  Vol.  64,  p.  715. 

2.  Wall  Across  Marsh,  on  Piles,  Engineering  Record,  Vol.  61,  p.  242. 

3.  Wall  on  Piles,  Engineering  Record,  Vol.  66,  p.  132. 

4.  Heavy  Gravity  Section,  Railway  Improvement,  Engineering  Record, 
Vol.  66,  p.  720. 

5.  Wall,  33  Feet  High,  on  Piles,  Journal  Western  Society  of  Eng.,  Vol. 
16  (1911),  p.  970. 

Walls  to  Meet  Special  Conditions 

1.  Retaining  Wall  as  Beam  over  Arch,  Engineering  Record,  Vol.  64,  p.  715. 

2.  Raising  Existing  Wall  (see  Fig.  40),  Journal  W.  S.  E.,  Vol.  16,  p.  970. 


78  RETAINING  WALLS 

Section  avoided  the  necessity  of  deep  excavation,  with  consequent  heavy 
shoring  of  adjacent  tracks.  The  abutting  piivate  property  made  it  impos- 
sible to  place  face  forms  for  a  concrete  wall,  and  a  rubble  masonry  wall 
was  built  instead,  backed  by  concrete.  The  author  adds  an  interesting 
note:  "It  has  occurred  to  the  writer,  that  there  is  one  feature  of  this  type 
of  wall,  that  might  frequently  be  employed  as  a  measure  of  economy.  That 
is  the  saving  in  excavation  and  masonry  effected  by  setting  the  foundation 
of  the  heel  higher  than  the  foundation  of  the  toe.  There  are  usually  but 
two  reasons  for  carrying  the  foundations  of  a  retaining  wall  lower  than  the 
surface  of  the  ground.  The  first  is  to  reach  a  material  that  will  sustain  a 
greater  pressure  and  the  second,  to  get  the  foundation  below  the  action  of 
frost.  The  first  is  usually  only  necessary  at  the  toe  of  the  wall,  for  almost 
any  good  soil  will  sustain  the  heel  pressure.  The  second,  also,  is  only  neces- 
sary under  the  toe  for  the  heel  is  protected  from  frost  by  the  embankment." 


CHAPTER  III 
DESIGN  OF  REINFORCED  CONCRETE  WALLS 

General  Principles. — Reinforced  concrete  retaining  walls  form 
a  class  of  walls  in  which  the  weight  of  the  earth  sustained  is  the 
principal  force  in  the  stability  moment.  Typical  sections  of  this 
class  of  wall  are  shown  in  Fig.  41.  The  same  fundamental  prin- 
ciples governing  the  general  outlines  of  the  gravity  wall,  as  given 
in  the  preceding  chapter,  likewise  govern  the  outlines  of  this  type 
of  wall  and  the  same  criteria  against  impending  failure  must  be 
satisfied.  The  actual  section  of  the  wall,  once  the  forces  upon 
it  are  known,  is  determined  from  the  principles  of  design  of 
reinforced  concrete,  a  brief  outline  of  which  principles  is  given 
in  this  chapter. 

As  in  the  case  of  gravity  walls,  the  stress  system,  soil  pressures 
arid  other  wall  functions  are  known  only  when  the  final  section 


in 

'lT  Cantilever 


"Ttan+i  lever  Courrterfbrt 

FIG.  41. — Typical  reinforced  concrete  sections. 


of  wall  is  known.  This,  of  course,  necessitates  a  process  of 
trial  and  error  until  a  wall  section  has  been  found  satisfying 
most  economically  all  the  necessary  requirements  of  the  data  at 
hand.  On  the  other  hand,  assuming  the  standard  type  of  loading 
as  shown  in  Fig.  5  and  using  the  standard  thrust  equation  as 
given  in  (14),  and  adding  a  few  approximate  conditions,  a  ten- 
tative section  may  be  chosen  from  appropriate  tables,  varying 
but  little  from  the  final  section  of  wall. 

79 


80  RETAINING  WALLS 

Preliminary  Section. — The  masonry  composing  the  wall  proper 
of  a  reinforced  concrete  section  plays  but  a  minor  role  in  control- 
ling the  final  wall  section.  The  difference  in  its  weight  and  the 
weight  of  the  earth  retained  may  thus  be  ignored  and  a  skeleton 
section  of  wall  treated  as  shown  in  Fig.  42.  The  thickness  of 
the  vertical  arm  of  the  wall  is  that  demanded  by  the  stresses 
existing  within  it  (for  a  certain  minimum  thickness  because  of 
construction  limitations,  see  the  following  pages)  and  whatever 
batter  is  given  to  the  back  of  the  arm  is  that  necessary  to  take 
care  of  the  increasing  moments  and  shears  in  going  toward  the 
base  of  the  wall.  This  is  comparatively  a  small  batter,  and  for 
a  tentative  design  may  be  ignored.  The 
back  of  the  wall  is  then  taken  vertical  and 
the  thrust  upon  it  is  assumed  to  have  a 
horizontal  direction.  The  value  of  the 
earth  pressure  coefficient  J  is,  for  this 
condition  %  (see  Table  1). 

The  required  outline  of  the  wall  is  satis- 
factorily determined  when  the  ratio  be- 
tween the  width  of  base  and  height  of  wall 
FIG.  42. — Skeleton  wall,  is  known.  This  ratio  is  denoted  in  the 
following  work  by  k.  Controlling  the  de- 
termination of  this  ratio  are  the  location  of  the  point  of 
application  of  the  resultant  pressure,  the  toe  extension,  if  any 
is  assumed,  the  maximum  permissible  intensity  of  pressure  upon 
the  soil  at  the  toe  and  the  factor  of  safety.  The  value  of  the 
determination  of  this  factor  has  been  discussed  on  page  57. 

The  approximate  assumption  as  to  a  skeleton  outline  of  wall 
in  addition  to  the  adoption  of  the  standard  forms  of  loading 
and  thrust  makes  it  possible  to  determine  directly  the  value  of 
the  ratio  k  depending  upon  the  various  functions  enumerated 
above.  While  this  section  is  not  to  be  taken  as  the  final  one, 
it  is  sufficiently  correct  a  section  upon  which  to  base  estimates 
of  cost  and  to  determine  the  limitations  of  the  various  types  of 
the  walls  to  the  peculiar  conditions  at  hand. 

Based  upon  the  above  assumptions  the  following  relations 
between  the  various  criteria  affecting  the  wall  section  are  found. 
Refer  to  Fig  42.  This  is  known  as  the  "T"  type  cantilever 
wall  and  is  together  with  its  modified  "L"  shape  wall,  the  type 
of  most  frequent  occurrence.  The  thrust  !Fis  found  from  equation 
(14)  and  is  located  at  a  distance  Bh  above  the  base,  where  B  has 


REINFORCED  CONCRETE  WALLS  81 

been  defined  by  equation  (12)  and  may  be  found  from  Table  3. 
The  moment  of  the  thrust  about  the  toe  P  is  then 

TBh 

and  if  these  quantities  are  replaced  by  their  values  as  taken  from 
the  equation  mentioned,  the  thrust  moment  is 


=      Jg(l  +  3c)A3  (80) 

as  before  g  is  the  unit  weight  of  the  retained  earth,  and  is 
ordinarily  taken  as  100  pounds  per  cubic  foot. 
The  stability  moment  of  the  wall,  Ms  is 

Ms  =  Gy  (81) 

Since,  as  per  the  adopted  approximation,  the  difference  in  weight 
between  the  masonry  comprising  the  wall  and  the  weight  of  the 
retained  fill  is  ignored,  the  value  of  G  is 

G=  gh  (l-+c)  w  (I  -  i)  (82) 

i  is  the  ratio  between  the  length  of  the  toe  extension  and  the 
entire  width  of  the  base.  The  value  of  the  lever  arm  is 

'       ;     v_B_«_«<l+3      '          (83) 


Let  the  ratio  between  the  width  of  base,  w,  and  the  height  of 
wall,  h,  be  denoted  by  k.  If  the  factor  of  safety  of  the  wall  is 
taken  to  mean  the  ratio  between  stability  moment  and  the 
overturning  moment,  and  is  denoted  by  n, 

Ms  =  nM0  (84) 

From  (81),  (82)  and  (83) 

M8  =  *  gk*(l  +  c)(l  -  *2)  /i3  (85) 

From  equations  80,  84,  85, 

and  finally 


k  -    I     J(l  +  3c)n  ttWrt 

k  ~^3(l  +  c)(l-t»)  (86) 


82  RETAINING  WALLS 

expressing  the  ratio  between  the  width  of  the  base  and  the  height 
of  the  wall  in  terms  of  factor  of  safety  assumed  and  the  width  of 
toe  extension.  The  surcharge  ratio  c  and  the  earth  pressure 
coefficient  J  are,  for  the  purposes  of  the  problem,  independent 
of  the  functions  of  the  wall  outlines. 

To  establish  the  base  ratio  k  in  terms  of  the  location  of  the 
point  of  application  of  the  resultant  and  the  toe  extension  (and 
these  are  the  two  functions  generally  known,  or  easily  found 
in  advance),  take  moments  about  the  point  of  application  of  the 
resultant.  M0,  the  thrust  moment  remains  the  same  as  before 
and  is  given  by  equation  (80).  The  new  stability  moment  Ms 
is  related  to  that  found  in  equation  (81)  in  the  ratio  of  the  respec- 
tive lever  arms  of  the  force  G,  or  if  M'8  denotes  the  new  stability 
moment 


M\IM,  -  -f — r-  =  i  -  ,-£-,  (87) 

1  -[-  1  1  -f-  Z 

2 

Taking  moments  about  the  point  0,  M's  =  M0  and  from  (87)  and 
since  Ms  =  nM0 

.1^*'n  (88) 


A  relation  between  the  factor  of  safety,  the  location  of  the  point 
of  application  of  the  resultant  and  the  toe  extension  ratio. 
Inserting  this  value  of  n  in  (86) 


*  =  *te+^wfeg          (89) 


which  may  be  written 


_  3e)  (90) 

-  - 


Inspecting  this  last  expression,  it  is  seen  that  k  is  a  minimum 
when  the  factor  (i  —  e)  in  the  denominator  vanishes,  or  f  or  i  =  e. 

For  a  given  location  of  the  resultant  pressure  the  most  economical 
width  of  base  is  had  when  the  vertical  arm  is  placed  over  the  assumed 
point  of  application  of  the  resultant  pressure. 

When  the  back  of  the  wall  is  vertical,  as  is  assumed  in  the 
present  analysis,  /  has  the  value  %,  which  should  be  inserted 
in  expressions  (86)  and  (90).  Again,  introducing  this  value  of 


REINFORCED  CONCRETE  WALLS  83 

J   and   also,   the  economical   criterion  established   above   (90) 
becomes 


*  =    r™e 

The  application  of  these  equations  to  specific  problems  is 
shown  at  the  end  of  the  chapter. 

Distribution  of  Base  Pressures.  —  The  manner  of  the  distribu- 
tion of  pressure  on  the  base  is  again  controlled  by  the  type  of  soil 
upon  which  the  wall  will  rest,  with  an  advantage  over  the  gravity 
type  of  wall  in  that,  any  tension  developed  in  the  wall  may  be 
taken  care  of  by  proper  reinforcement.  Continuing  the  approxi- 
mations given  above,  further  guidance  may  be  had  in  shaping 
the  wall  to  meet  the  anticipated  soil  conditions. 

The  total  load  upon  the  base  of  the  wall  is  G.  From  (39)  of 
Chapter  11  and  from  (82)  above 

I±^^(2-3«)         (92) 

Place  H  =  h  (I  +  c);  that  is,  H  is  the  total  depth  of  fill  plus  the 
depth  of  surcharge.  Solve  the  equation  for  e,  taking  the  unit 
weight  g  of  the  earth  as  100  pounds  per  cubic  foot  and  expressing 
both  this  weight  and  the  soil  pressure  intensity  Si  in  tons.  There 

6  =  3  ~  (93) 


When  the  maximum  soil  pressure  intensity  S  is  given  as  well  as 
the  toe  extension  ratio  i,  this  equation  may  be  used  to  locate  the 
point  of  application  of  the  resultant  pressure  upon  the  base. 
When  this  value  of  e  has  been  found,  equation  (90)  is  then  ap- 
plied to  find  the  value  of  the  base  width  ratio  k. 

Conversely  when  the  point  of  application  of  the  resultant  is 
assigned  (and  with  a  foundation  known  in  advance,  the  location 
of  the  point  of  application  of  the  resultant  is  usually  indicated) 
the  toe  extension  necessary  to  give  this  resultant  location  is 
found  from 


(2  -  3e)H 

If,  in  equation  (93),  i  is  put  equal  to  e  (the  economy  criterion), 
and  the  resulting  equation  is  solved  for  e 

sTT 

(95) 


84 


RETAINING  WALLS 


Under  the  above  conditions,  given  H  and  Si,  the  toe  extension 
ratio  i  is  determined  at  once.  The  conditions  under  which  the 
location  of  the  stem  is  governed  solely  by  the  economy  of  the 
wall  have  been  previously  touched  upon  (see  pages  42^44) 
and  will  be  discussed  in  more  detail  further  on.  Clearly,  if  no 
limitation  is  placed  upon  the  location  of  the  vertical  arm,  it 
should  be  placed  where  the  economy  criterion  dictates :  directly 
over  the  indicated  position  of  the  point  of  application  of  the 
resultant  upon  the  base. 

Tables  and  Their  Use. — Tables  are  readily  founded  upon  the 
preceding  equations  and  simplify  the  necessary  calculation  of 
the  wall  outlines.  From  the  relation  existing  between  the  loca- 
tion of  the  point  of  application  of  the  resultant,  the  factor  of 

TABLE  17. — VALUES  OF  e 


~--^7l 

i  i--* 

2 

345 

0 

.25 

.33 

.38 

.40 

.1 

.27 

.37 

.41 

.44 

.2 

.30 

.40 

.45 

.48 

.3 

.33 

.43 

.49 

.4 

.35 

.46 

.5 

.38 

.50 

safety  and  the  amount  of  toe  projection,  equation  (88),  Table 
No.  17  has  been  prepared.  With  a  given  location  of  the  resultant 
and  an  assigned  factor  of  safety,  the  required  toe  projection  is 
taken  from  the  table.  Again,  for  an  assigned  location  of  the 
point  of  application  of  the  resultant  and  a  given  toe  projection, 
the  factor  of  safety  may  be  taken  from  the  same  table.  For 
the  criterion  of  economy  i.e.  i  =  e,  this  relation  becomes 


n  -  1 


e  = 


n+  1 


n 


I  -  e 


(96) 


TABLE  18. — VALUES  OF  k 




c 

T-l 
« 

(N 

eo 

• 

i 

» 

II 

i 

• 

»' 

II 

• 

• 

• 

• 

n 

•' 

j 

• 

0 

.37 

.43 

.53 

.74 

.37 

.42 

.48 

.57 

.77 

.38 

.42 

•H 

.56 

.70 

.41 

.45 

.50 

.56 

.67 

.1 

.42 

.49 

.(•() 

.85 

.43 

.48 

.54 

.65 

.88 

.44 

.48 

.54 

.64 

.81 

.47 

.51 

.57 

.64 

.76 

.4 

.47 

.54 

.60 

.94 

.47 

.53 

.60 

.72 

.97 

.48 

.53 

.60| 

.70 

.89 

.52 

.57 

.63 

.71 

.84 

REINFORCED  CONCRETE  WALLS  85 

A  general  table,  Table  18  has  been  prepared,  giving  the  value 
of  k,  as  found  from  equation  90,  for  a  range  of  values  of  c,  e  and  i. 
The  earth  pressure  constant  J,  has  been  taken  as  %. 

With  the  general  outlines  of  the  wall  approximately  established 
by  aid  of  the  foregoing,  it  is  possible  to  proceed  with  the  actual 
design  of  the  several  members  composing  the  reinforced  concrete 
retaining  wall.  While  it  is  not  the  purpose  of  the  preceding 
analysis  to  replace  a  careful,  exact  analysis  of  the  wall,  its  prime 
intent  is  to  permit  an  intelligent  selection  of  a  wall  without  a 
tedious  process  of  trial  and  error.  It  should  be  pointed  out,  that 
the  approximations  consist  in  ignoring  factors  which  have  proven 
negligible  in  controlling  the  wall  dimensions,  so  that  even  though 
the  selection  of  the  wall  outlines  are  finally  determined  by  these 
approximations,  no  serious  error  has  been  committed.  However, 
a  careful  and  painstaking  designer  will  analyze  the  completed 
wall,  to  see  whether  the  stress  system  in  it  checks  with  the  one 
first  determined. 

Theory  of  the  Action  of  Reinforced  Concrete.— The  assump- 
tions in  the  design  of  reinforced  concrete  beams  are  those  of  the 
ordinary  beam  theory,  namely:  the  Bernoulli — Euler  theory  of 
flexure.  The  fundamental  premise  is  that  a  plane  section  before 
bending,  remains  a  plane  section  after  bending,  with  the  further 
assumption  that  Hooke's  Law,  i.e.  the  stress  is  proportional  to 
the  strain,  is  true. 

Although  the  brilliant  researches  of  Barre  de  St.  Venant, 
have  shown  that  plane  sections  do  not  remain  plane  during  bend- 
ing, the  error  becomes  appreciable  when  the  ratio  of  depth  of 
beam  to  span  exceeds  one-fifth.  Since  for  such  ratios,  stresses, 
other  than  those  induced  by  bending  moment,  usually  govern 
the  required  reinforcement  and  depth  of  beam  e.g.  the  unit  shear 
and  adhesion,  these  assumptions  of  plane  sections  may  be  taken 
as  valid,  so  long  as  the  stresses  induced  by  the  bending  moment 
govern  the  required  depths  and  amounts  of  steel  reinforcement. 
The  concrete  is  assumed  to  take  no  tension. 

The  excellent  report  of  the  Special  Committee  on  Concrete 
of  the  A.S.C.E.,  has  set  the  seal  of  approval  on  this  mode  of 
figuring  the  action  of  reinforced  concrete  after  most  thorough 
investigation,  both  from  a  theoretical  and  experimental  stand- 
point, and  the  engineer  may  accept  this  method,  with  no  fear  of 
beam  failure  ensuing,  so  long  as  care  has  been  taken  of  all  the 
stress  criteria. 


86 


RETAINING  WALLS 


Under  load,  the  distribution  of  stress  across  a  section  normal 
to  the  axis  of  the  beam  is  shown  in  Fig.  43.  Adopting  the  recom- 
mended nomenclature  as  suggested  in  the  above  report,  E8  is 
the  steel  modulus,  Ec  the  concrete  modulus,  and  n  the  ratio  of 

the  steel  modulus  to  the 

b  /ec          fc^bd       concrete  modulus.     A8  and 

Ac  are  the  areas  of  the  steel 
and  concrete  in  the  section 
respectively.  fc  and  fs  are, 
respectively,  the  unit  con- 

FIG.  43.— Theory  of  reinforced  concrete.  crete     and     steel     Stresses. 

Let  ec  be  the  displacement 

of  the  section  at  a  and  e8  that  at  b.  From  the  assumption  that 
a  plane  section  remains  a  plane  section  after  bending,  and  from 
Hooke's  Law 


c----t 

0         0 

a        a  la'           fc  / 

if  :« 

^T 

^ 

;d  J 

~-i.JL  

kd 


fc= 


(l-k)d      I  -  k 

=  e8E8As]  As  =  pAc  =  pbd 


(97) 


and,  by  summation  of  all  the  horizontal  forces 

fe  kbd  Ececkbd 

JJL        =  e8E8A8)  or  — -| —  =  e8E8pbd 

-  & 

kecEc 


whence 


and  equating  this  to  (97) 


ec  _  2pn 
eg        k 


and  finally 


Solving  this  for  k 


2pn          k 
k      =  1  -k 

kz  +  2kpn  —  2pn  =  0. 


(98) 


(99) 


k  = 


+  2pn  -  pn 


(100) 


which  locates  the  position  of  the  neutral  axis,  once  the  ratio  of 

the  two  moduli  are  adopted  and  the  percentage  of  steel  assumed. 

It  is  to  be  noticed  that  it  is  a  function  of  these  two  quantities  only. 

The  resisting  moment  of  the  section  may  be  expressed  with 


REINFORCED  CONCRETE  WALLS  87 

either  the  steel  force  or  the  concrete  force  as  the  force  factor  in 
the  couple.  If  Mc  and  Ms  are  the  concrete  and  steel  resisting 
moments  respectively, 

~  o 


';       M.  =  f,A,  l-d-  f,p   l  - 

1  —  k/3  =  j  and  is  the  effective  lever  arm  of  the  couple,  corre- 
sponding to  the  effective  depth  of  homogeneous  beams.  The 
moments  may  be  expressed  as 

Mc  =  kcbdz;Ms  =  kabd*  (101) 

where 

ke  =  fckj/2;  ks  =  fspj  (102) 

Ordinarily,  the  most  economical  section  is  that  one  in  which 
the  concrete  and  the  steel  are  each  stressed  to  their  permissible 
limits.  The  percentage  of  steel  to  satisfy  this  condition  may  be 
found  as  follows  : 

Since,  from  the  summation  of  horizontal  components  of  stress 
intensities  across  the  right  section  of  the  beam,  the  total  concrete 
stress  must  be  equal  to  the  total  steel  stress 

A8fs  =  pbdfs  =  kbdfc/2 
from  which  equality 

k  =  2p£  (103) 

Jc 

Equating  this  value  of  k  to  that  found  in  equation  (100)  and 
replacing  the  ratio  2fs/fe  by  a,  and  solving  the  equation  for  p 

;  =  -  __  ?*__       _??_  ao4) 

(n  +  a)2  -  n*       a(2n  +  a) 

If  in  the  ratio  a,  the  unit  stresses  are  those  allowed  for  the 
material  at  hand,  than  this  value  of  p  proves  to  be  the  most 
economical  one  to  use. 

The  above  analysis  is  of  course,  predicated  upon  the  assump- 
tion that  the  section  is  controlled  by  the  bending  moment.  Other 
stresses  may  determine  the  percentage  of  steel  or  the  depth 
of  the  section.  When  the  percentage  of  steel  is  above  that  neces- 
sary for  the  economical  steel  ratio  as  given  by  (104),  then  the 
concrete  stress  in  the  section  will  determine  the  resisting  mo- 
ment to  be  used  and  the  section  constant  is  found  from  kc,  as 


88 


RETAINING  WALLS 


defined  above.  With  this  value  of  kc,  the  proper  percentage 
of  steel  is  to  be  taken  from  Table  19.  Again  the  depth  of  the 
section  may  be  greater  than  required  by  the  bending  moment,  and 
accordingly  the  percentage  of  steel  to  satisfy  the  bending  moment 
will  be  less  than  that  required  by  equation  (104).  The  steel 
stress  will  be  the  governing  stress  in  the  section  and  the  section 
constant  to  be  used  will  be  ks  as  defined  in  equation  ( 102) .  The 
proper  percentage  is  found  from  Table  19  with  this  value  of  ks. 
The  conditions  under  which  these  constants  control  are  best 
illustrated  by  specific  problems  as  given  at  the  end  of  the  chapter. 


O.OOZ     0.003     0.004-    0.005 


O.OOd  0.009    0.010 


0.006  0.007 
_p= Steel  Ratio  ^ 

Curve  Plate  No.  2. 
Economical  steel  percentage. 


To  simplify  the  use  of  equation  (104),  Curve  Plate  No.  2  has 
been  drawn  from  which  the  proper  value  of  p  may  be  taken  once 
the  value  of  n  and  of  the  ratio  a  are  known. 

In  the  report  of  the  Special  Concrete  Committee,  mentioned 
above,  the  following  values  of  n  are  suggested,  depending  upon 
the  ultimate  strengths  of  concrete: 

n  =  15.  Ultimate  strength  equal  to  or  less  than  2200  Ibs.  per  sq.  in. 
n  =  12.  Ultimate  strength  between  2200  and  2900  Ibs.  per  sq.  in. 
n  =  10.  Ultimate  strength  greater  than  2900  Ibs.  per  sq.  in. 

Table  No.  19  is  a  compilation  of  the  values  of  the  several 
functions  entering  into  the  computation  of  a  concrete-stee 
section.  It  is  noticed  that  the  terms  are  not  carried  out  to  the 
usual  degree  of  refinement.  In  view  of  the  approximation  in 


REINFORCED  CONCRETE  WALLS 


89 


both  the  theory  and  in  the  experimental  determination  of  the 
concrete  constants,  it  does  not  seem  good  practice  to  carry  the 
work  out  to  any  greater  degree  of  exactness  than  shown  here. 

TABLE  19. — REINFORCED  CONCRETE  CONSTANTS 


n  =  10 

n  - 

•1' 

=  12 

H# 

PJ 

n  =  15 

P 

k 

j 

Hik 

PJ 

ft 

j 

VzJk 

pj 

.002 

.18 

.94 

.09 

.002 

.20 

.93 

.09 

.002 

.22 

.93 

.10 

.002 

.004 

.25 

.92 

.12 

.004 

.26 

.91 

.12 

.004 

.29 

.90 

.13 

.004 

.006 

.29 

.90 

.13 

.005 

.31 

.90 

.14 

.005 

.34 

.89 

.15 

.005 

.008 

.33 

.89 

.14 

.007 

.35 

.88 

.15 

.007 

.38 

.87 

.17 

.007 

.010 

.36 

.88 

.16 

.009 

.38 

.87 

.16 

.009 

.42 

.86 

.18 

.009 

.012 

.38 

.87 

.17 

.010 

.41 

.86 

.18 

.010 

.45 

.85 

.19 

.010 

.014 

.40 

.87 

.17 

.012 

.44 

.85 

.19 

.012 

.47 

.84 

.20 

.012 

.016 

.43 

.86 

.19 

.014 

.46 

.85 

.20 

.014 

.49 

.84 

.21 

.013 

.018 

.45 

.85 

.19 

.015 

.48 

.84 

.20 

.015 

.51 

.83 

.21 

.015 

.020 

.47 

.85 

.20 

.017 

.49 

.84 

.21 

.017 

.53 

.82 

.22 

.016 

.025 

.50 

.83 

.21 

.021 

.53 

.82 

.22 

.020 

.57 

.81 

.23 

.020 

.030 

.53 

.82 

.22 

.025 

.56 

.81 

.23 

.025 

.60 

.80 

.24 

.024 

In  addition  to  determining  the  resisting  moment  of  a  section, 
it  is  necessary  to  find  the  unit  shear  and  the  unit  adhesion,  each 
of  which  stresses  may  demand  more  resisting  material  than  that 
required  by  the  moment. 

Analagous  to  a  steel  or  other  section  of  homogeneous  material 
the  shear  over  any  section  is  assumed  distributed  over  the  effec- 
tive depth  (jd)  of  the  section,  so  that,  if  s  is  this  unit  shear,  and 
V  is  the  total  shear  over  the  section 


V 

jbd 


(105) 


The  unit  adhesion  corresponds  again,  to  the  horizontal  shear, 
and  since  the  unit  vertical  shear  is  equal  to  the  unit  horizontal 
shear,  the  periphery  of  the  steel  embedded  in  the  concrete  per 
unit  length  must  carry  the  unit  horizontal  shear  (or  its  equiva- 
lent, the  unit  vertical  shear.) 

If  r  is  the  periphery  of  the  rods  per  unit  length,  and  q  is  the 
permissible  adhesion  stress, 


_ 

jdr 


(106) 


90  RETAINING  WALLS 

TABLE  20. — STANDARD  ULTIMATE  STRENGTHS  OP  AGGREGATES  AS  SUGGESTED 
BY  THE  SPECIAL  COMMITTEE  ON  CONCRETE  A.  S.  C.  E. 


Aggregate 

1  :1:2 

1:1K:3 

1  :2:4 

1:2^:5 

1  :3:6 

Granite,  trap       .        .    . 

3300 

2800 

2200 

1800 

1400 

Gravel,  limestone  
Soft  limestone  

3000 
2200 

2500 
1800 

2000 
1500 

1600 
1200 

1300 
1000 

Cinders 

800 

700 

600 

500 

400 

The  following  are  the  percentages  of  the  above  ultimate  stresses  that  may 
be  allowed: 

Bearing. — Compression  applied  to  surface  twice  the  loaded  area,  32.5 
per  cent. 

Axial  Compression. — Where  the  length  is  not  greater  than  twelve  diame- 
ters, 22.5  per  cent. 

Compression  Extreme  Fibre. — 32.5  per  cent. 

Shear  and  Diagonal  Tension. — Beams,  with  horizontal  bars  (i.e.,  bars 
parallel  to  the  longitudinal  axis  of  the  beam  only)  no  web  reinforcement, 
2  per  cent. 

Bond. — 4  per  cent.     In  case  of  wires  2  per  cent. 

Upon  the  recommendation  of  the  above  Committee,  Table  20 
was  compiled,  giving  the  standard  ultimate  strengths  for  the 
several  combinations  of  the  different  aggregates,  and  then  the 
percentages  of  these  ultimate  loads  to  be  used  for  the  different 
type  of  stresses. 

Bending  and  Anchoring  Rods. — Rods  are  anchored  in  the  con- 
crete by  (1)  carrying  them  beyond  the  theoretical  end,  a  distance 
sufficient  to  develop,  in  bond,  its  tensile  stress;  (2)  hooking  the 
end  of  the  rod  around  a  rod  at  right  angles  to  it;  (3)  threading 
the  end  of  the  rod  and  bolting  it  to  a  steel  washer  or  other  steel 
device  buried  in  the  concrete  (4)  making  a  U  turn  in  the  rod. 
The  first  and  last  methods  are  the  usual  ones  because  of  cheap- 
ness of  these  details.  The  second  and  third  are  used  only  where 
lack  of  room  makes  such  details  necessary.  Bending  rods  around 
another  rod,  and  threading  and  bolting  rods  are  expensive  details 
to  be  avoided  as  far  as  possible. 

If  the  unit  adhesion  is  q  and  fs  is  the  steel  stress,  then,  if  L  is 
the  length  necessary  to  carry  the  rod  beyond  its  theoretical  end 

4qtL  =  fat2  and  L  =  |s  t.  (107) 

The  value  of  this  fraction  varies  from  40  to  50  (the  unit  stresses 
taken  from  Table  20)  and  the  rod  is  carried  passed  the  theoretical 
end,  this  number  of  thicknesses. 


REINFORCED  CONCRETE  WALLS  91 

If  a  rod  is  twisted  about  another  rod  then  the  twist  should  be 
at  least  one  complete  turn  (360°)  and  carried  beyond  about 
six  inches,  not  only  to  satisfy  the  theoretical  requirements,  but 
to  aid  the  work  in  the  field.  In  bending  a  rod  care  must  be  taken 
that  the  radius  to  which  the  rod  is  bent  is  sufficiently  large  that 
the  bearing  induced  on  the  concrete  will  be  within  the  allowable 
limits.  For  a  rod  bent  to  a  circular  arc,  carrying  a  tension  of  T 
at  either  end,  the  condition  is  similar  to  that  of  a  hoop  (see  any  text 
on  applied  mechanics)  and  the  compressive  stress  upon  the  concrete 
per  linear  unit  of  the  rod  is 

C  =  T/R 

Where  R  is  the  radius  of  the  bend.  If  c  is  the  permissible  unit 
bearing  on  concrete  and/s  is  the  permissible  steel  unit  stress,  then 
introducing  these  factors  in  this  last  equation 


t  is  the  thickness  of  the  rod.  The  ratio  fs/c  has  a  value  of  about 
30  and  in  the  work  that  follows  this  proportion  will  be  used  in 
determining  the  proper  radius  to  turn  the  rod. 

To  get  the  area  of  a  washer  necessary  to  hold  the  bar,  with  A 
the  area  of  the  washer  and  c  the  unit  concrete  bearing,  let  d 
be  the  side  of  the  square  (if  a  square  washer  be  used)  and  with  the 
same  units  as  before,  the  total  bearing  is  Ac.  Since  Ac  = 


+V  (108) 

With  the  usual  unit  stresses,  d  is  about  six  thicknesses  of  the  bar. 
If  d  is  the  diameter  of  a  round  washer 


d-ttv/ftt2Ifl  (109) 


With  the  usual  values,  the  diameter  of  a  round  washer  should  be 
about  seven  and  one-half  thicknesses  of  the  bar. 

Vertical  Arm. — The  vertical  arm  of  a  reinforced  concrete  wallas 
shown  in  Fig.  42  and  as  tentatively  analyzed  on  pages  80  and  82  is 
a  cantilever  beam,  subjected  to  a  horizontal  load  of  T,  located  at 
a  point  Bh  above  the  base.  In  the  skeleton  wall,  the  basis  for  the 
approximate  analysis,  h  is  measured  from  the  bottom  of  the  wall. 
In  the  actual  final  section,  the  correct  value  of  h  must  be  used, 


92  RETAINING  WALLS 

namely  the  height  of  the  vertical  wall  above  the  top  of  the  foot- 
ing. The  discrepancy  in  the  assumed  and  correct  h  may  be 
ignored  in  the  tentative  selection  of  the  thicknesses  of  the  arm 
and  footing. 

As  above  shown  the  cantilever  moment  in  the  arm  is  TBh,  and 
if  T  is  replaced  by  its  value  in  (14),  and  B  by  its-  value  in  (12) 
then 

M  =\Jg(l  +  3c)ft3  (110) 

The  value  of  J  is  taken  as  one-third  (see  page  80).  g  is 
the  unit  weight  of  earth  and  c  is  the  ratio  of  the  surcharge  height 
to  the  actual  height  of  wall  assumed.  The  standard  type  of 
loading  as  shown  in  Fig.  5  is  to  be  used. 

While  the  shear  and  the  unit  adhesion  may,  and  frequently 
do,  control  the  depth  of  beam  required,  this  depth  will  not  vary 
much  from  that  required  by  the  bending  moment  depth  and 
it  is  safe  in  this  preliminary  analysis  to  work  with  the  depth 
required  by  the  bending  moment.  The  resisting  moment  has 
been  given  by  (101)  and  equating  this  to  the  external  moment 
given  in  (110),  and  solving  for  d 


=  AN-^r 

J  may  be  given  the  value  J^  as  above,  g  is  taken  at  the  usual 
weight  100  Ib.  per  cubic  foot.  If  the  economy  criterion  of 
(104)  is  used,  and  if  in  accordance  with  general  practice  a  1:2:4 
concrete  is  specified  with  the  resulting  permissible  stresses  as 
given  in  Table  20,  from  Curve  Plate  No.  2  with  n  =  15,  the  steel 
ratio  p  is  0.0075.  From  Table  19,  0.5&?  is,  for  this  value  of  p, 
0.17  and  since  /«.,  in  conformity  with  the  other  terms  of  (111)  is 
to  be  expressed  in  units  of  pounds  per  square  foot,  the  bending 
moment  constant  kc  from  equation  (102)  is  about  16,000. 
With  these  values  equation  (111)  becomes 


d  =  0.0185tf*V(T+"3c)  (H2) 

The  depth  d  necessary  to  satisfy  the  bending  moment  due  to 
the  earth  thrust  may  be  closely  approximated  from  this  equation 
and  the  same  expression  may  be  used  to  find  the  required  depth 
at  any  point  on  the  cantilever  arm,  by  using  the  proper  values  of 
c  and  h. 


REINFORCED  CONCRETE  WALLS  93 

To  determine  the  depth  to  satisfy  the  shear  requirements,  ap- 
ply equation  (105).  V  is  the  thrust  T  and  j  may  be  safely 
taken,  for  the  purposes  at  hand,  at  %.  With  the  same  concrete 
constants  as  assumed  above,  the  shearing  value  for  a  simply  rein- 
forced beam  is  s  =  40  pounds  per  square  inch  or  5760  pounds  per 
square  foot.  The  required  value  of  d  is 

d  =  r/5040  =  Jgh2(l  +  2c)/10080  =  0.0033A2(1  +  2c)   (113) 

Comparing  this  equation  with  (112),  the  shearing  stress  will 
control  the  required  depth  of  the  arm,  whenever 
TABLE  21       the  value  of  d  as  found  from  (113)  is  greater  than 
hl        that  value   as  found  from   (112).      Solving    this 
—     inequality,  the  shearing  stress  will  determine  the 


.0 
.1 
.2 
.3 
.4 
.5 
.6 
.7 
.8 
.9 
1.0 


31        necessary  depth  when 

28 

25  h>    n 

23  (1  + 

21 


This  may  be  termed  the  "critical"  value  of  h 
18  and  Table  21  gives  the  values  of  the  "critical" 
17  value  of  h  for  several  values  of  surcharge  ratio  c. 
16  Its  use  is  explained  in  the  problems  at  the  end  of 
the  chapter.  The  above  equations  suffice  to  de- 
termine, approximately,  the  thickness  of  the  arm 
to  satisfy  the  stresses  induced  by  the  earth  thrusts. 
While  such  thicknesses  are  fairly  accurate  (the  problems  at  the 
end  of  the  chapter  are  illustrative  of  this)  it  is 
better  practice  to  take  the  wall  thus  approxi- 
mately outlined  as  the  tentative  section  and 
design  finally  by  the  more  exact  methods  the 
required  dimensions  of  the  wall. 

Footing.—  The  footing,  see  Fig.  44,  is  again  a      | 
cantilever,   with    its  maximum   moment   at  the 
foot  of  the  vertical  arm  B.      Its  loading  is  the     ^g  on  fating." 
net  difference  between  the  downward  weight  of 
the  retained  fill  and  the  upward  thrust  of  the  soil  pressures.     The 
soil  pressure  intensity  at  B  is 


(1  -  t)Si  +  iS2     (115) 
Taking  moments  at  B 

MB  =  Gp/2  -  S2p*/2  -  (SB  -  S2)pV6  =  Gp/2  -  (28,  +  SB) 
P2/6  (116) 


94  RETAINING  WALLS 

From     (115) 

2S2  +  SB  =  (1  -  i)  S}  +  (2  +  i)  S2 
and  from  (39)  and  (40)  of  chapter  2 

2S2  +  SB  =  Q^[e  -  i(l  -  26)] 
The  expression  (116)  for  the  bending  moment  now  becomes 


(H7) 

Note  that  p  =  w(l  —  i)  and  that 

G  =  gh(l  +  c)  (1  —  j)w;     and     w  =  kh 
Using  the  value  of  k  as  found  in  equation  (90),  the  expression 


for  the  bending  moment  (117)  is  finally 
MB  = 

where,          /  =  — 


(118) 
(119) 


Comparing  the  value  of  this  moment  as  given  in  equation  with 
that  of  the  vertical  arm,  as  given  in  equation  (110),  it  is  seen  that 
the  footing  moment  is  /  times  the  arm  moment  with  I  varying 
from  one  to  one-half.  Table  22  gives  a  series  of  values  of  7. 

TABLE  22 


i 

e  =  0 

e  =  0.4 

e  =  0.5 

I 

Q 

I 

Q 

I 

Q 

.0 

1.00 

.00 

1.00 

.00 

1.00 

.00 

.1 

.96 

.03 

.95 

.05 

.90 

.10 

.2 

.88 

.11 

.85 

.14 

.80 

.20 

.3 

.76 

.19 

.72 

.25 

.70 

.30 

.33 

.72 

.22 

.69 

.28 

.60 

.33 

.4 

.64 

.27          .62 

.34 

.60 

.40 

.5 

.50 

.33   !       .50 

.43 

.50 

.50 

As  before,  the  shearing  stresses  and  the  adhesion  stresses  must 
be  found.  The  complicated  type  of  loading  upon  the  footing 
makes  it  impossible  to  find  an  easily  applied  expression  for  these 
stresses  and  resort  must  be  had  to  specific  problems  to  illustrate 


REINFORCED  CONCRETE  WALLS 


95 


the  effects  of  these  stresses.     Some  problems  at  the  end  of  this 
chapter  bring  out  in  detail  these  points. 

Toe  Extension. — The  approximate  design  of  the  toe  extension 
of  the  footing,  if  such  an  extension  is  used,  follows  along  lines 
similar  to  those  of  the  preceding  paragraphs.  Referring  again 
to  Fig.  44  with  the  value  of  the  soil  intensities  as  previously  found 
SB  is  taken  the  same  as  in  the  design  of  the  heel  extension.  For 
the  exact  analysis,  the  moments  for  the  heel  and  the  toe  are  taken 
at  the  intersection  of  the  rear  and  face  planes  of  the  vertical  arm 
respectively.  For  the  approximate  solutions  now  sought  this 
refinement  is  unnecessary  and  taking  moments  about  B 

rr  rr 

919   /O         I          *^1  *^B       *97   91. 


M'B    =   S 


(8, 


(120) 


and  again  replacing  the  soil  intensities  and  k  by  their  values, 


where 


-2s)] 


(121) 


(122) 


The  toe  footing  moment  is  thus  Q  times  the  arm  moment, 
with  Q  varying  from  zero  to  one-half.  Table  22  gives  a  set  of 
values  for  Q. 


FIG.  45. — Graphical  analysis  of  reinforced  concrete  wall. 

It  is  again  necessary  to  emphasize  the  fact  that  the  shearing 
and  adhesion  stresses  must  be  ascertained. 

The  dimensions  of  the  wall  are  thus  approximately  determined, 
and  with  the  outlines  of  the  wall  previously  found,  it  is  possible  to 
proceed  with  the  definite  final  design.  Laying  out  the  wall  in 


RETAINING  WALLS 


accordance  with  these  dimensions,  the  thrust  may  be  found  by 
the  graphical  methods  or  may,  once  more,  be  taken  with  J  = 
one-third  as  urged  in  Chapter  I  and  then  combined  with  the  ver- 
tical weight  of  the  earth  on  the  projection  of  the  back  of  the  arm 
(if  the  arm  be  battered  from  the  minimum  practical  width  at  the 
top  to  the  required  width  at  the  base).  With  the  thrust  deter- 
mined, the  location  of  the  resultant  and  the  soil  pressure  intensi- 
ties are  found  and  checked^with  the  location  and  intensities  of 
pressure  assumed  originally.  This  is  best  found  graphically  as 
shown  in  Fig.  45,  where  the  properties  of  the  funicular  polygon 
are  utilized.  Several  problems  at  the  end  of  this  chapter  develop 
in  greater  detail  the  methods  sketched  here. 

Counterfort  Walls. — -A  study  of  the  expressions  determining 
the  thicknesses  of  the  members  of  the  cantilever  walls  discussed 
in  the  preceding  sections,  will  show,  that  as  the  walls  increase 
in  height,  the  required  thicknesses  of  these  members  become 
very  large.  To  reduce  the  sizes  of  the  arm  and  of  the  footing, 
supporting  walls  are  introduced  between  these  members,  termed 
loosely,  counterforts.  See  Fig.  46.  These  serve  a  function  similar 

to  that  performed  by  the  gusset 
plate  on  a  through  girder,  an- 
choring the  wall  and  base  slab 
to  each  other. 

This  combination  of  counter- 
fort, wall  and  footing,  forms  a 
structure  quite  difficult  to 
analyze  exactly  and,  generally, 
no  such  exact  analysis  is  at- 
tempted. The  usual  modes  of 
treating  the  wall  and  base  slabs 
of  the  counterfort  wall  are  as 
follows : 

(a)  The  wall  and  the  footing 

are  treated  as  composed  of  a  series  of  independent 
longitudinal  strips,  freely  supported  at  the  ends,  i.e.,  at  the 
counterforts.  The  bending  moment  is  then  WL/S.  W  is  the 
total  weight  acting  upon  the  strip  in  question. 

(6)  The  wall  and  footing  are  treated  in  strips  as  above,  but  the 
supports  are  taken  as  fixed  at  the  counterforts.  Although,  ex- 
actly speaking,  for  this  condition,  the  moment  at  the  support  is 
WL/12,  and  that  at  the  center  of  the  beam  is  TFL/24,  the  moment 


FIG.  46. — Stresses  in  a  counter- 
forted  wall. 


REINFORCED  CONCRETE  WALLS  97 

is  assumed  alike  at  the  center  and  at  the  support  and  of  value 
WL/12. 

Method  (6)  is  the  one  generally  used  in  the  design  of  the  slabs 
forming  the  counterfort  wall  and  will  be  used  in  the  present  text. 

The  design  of  the  counterfort  itself  is  a  matter  of  much  con- 
troversy and  practice  is  far  from  uniform  here.1  It  may  be  taken 
as  a  tension  brace,  simply  anchoring,  by  means  of  the  rods  con- 
tained in  it,  the  base  slab  and  the  wall  slab  to  each  other,  the 
concrete  merely  acting  as  a  protection  to  the  steel;  as  a  cantilever 
beam,  anchored  at  the  base  and  receiving  its  load  from  the  wall 
slab,  or  as  the  stem  of  a  "T"  beam.  In  the  following  work  the 
counterfort  will  be  treated  as  a  cantilever  beam.  Prof.  Cain  has 
made  an  exact  analysis  of  a  beam  of  this  wedge  shape  (see  his 
"Earth  Pressures,"  etc.)  but  the  theory  of  retaining  walls  and  of 
earth  pressures  does  not  seem  to  justify  such  refinements  of 
design. 

Not  only  are  all  of  the  methods  of  stress  computation  above 
discussed  approximate,  but  it  is  difficult  to  make  an  estimate  as  to 
their  degree  of  exactness.  If  the  slabs  are  designed  as  outlined 
under  (a)  and  (b)  the  relieving  action  of  the  portion  of  the  slab 
adjacent  to  the  strip  under  question  is'  ignored.  That  is,  no 
account  is  taken  of  the  plate  action  that  may  exist  in  the  slab. 
Toward  the  junction  of  the  base  and  the  arm,  the  two  members 
tend  to  mutually  stay  each  other,  reducing  the  possible  deflec- 
tion and  thus  the  resulting  stress.  It  is  clear  that  there  is  con- 
siderable latitude  permissible  in  making  stress  assumptions  and 
here  again,  simplicity  of  design  should  dictate  the  formulas  to 
be  used  rather  than  an  intricate  analysis  of  questionable  accuracy. 

While  attention  has  been  paid  only  to  bending  moments  in 
discussing  stresses,  it  is  understood  that  the  other  stresses,  such 
as  shear  and  adhesion  are  likewise  to  be  ascertained,  and,  in 
fact,  it  will  be  seen  that  these  latter  stresses  may  more  often  con- 
trol the  required  dimensions  than  the  bending  moment  stress. 

Face  Slab. — -The  same  assumptions  as  to  standard  character 
of  loading,  of  amount  of  earth  thrust  etc.  will  obtain  here  as  have 
obtained  in  the  former  work  on  the  design  of  the  walls.  The 
intensity  of  earth  pressure  upon  any  horizontal  strip  (see  Fig.  46) 
at  a  depth  x  below  the  top  of  the  wall  is 

Jx(l  +  cjflf  <123) 

1  See  E.  GODFREY,  Trans.  A.S.C.E.,  Vol.  Ixx,  p.  57,  and  accompanying 
discussion. 

7 


98  RETAINING  WALLS 

where  J  is  to  be  taken  at  its  usual  value  J£  ;  cx  is  the  ratio  of  the 
surcharge  height  h'  to  x  and  g  is  the  unit  weight  of  the  earth. 
If  m  is  the  counterfort  spacing,  and  if  the  moment  is  as  above 
denned  TFL/12,  then 

M  =  x(l+  cx)0w2/36  (124) 

Placing  x  =  vh,  so  that  v  is  the  ratio  between  the  distance  from 
the  top  of  the  wall  to  the  point  in  question  and  the  total  height  of 
the  wall,  then  cx  =  h'/x  =  c/v;  where  c  is  the  standard  ratio 
between  the  surcharge  height  hr  and  the  total  height  h.  The  mo- 
ment may  now  be  placed 

M  =  h(c  +  v)  0m2/36  (125) 


As  before  (see  page  92),  the  resisting  moment  of  the  slab,  for  a 
condition  of  balanced  reinforcement  may  be  placed  equal  to  kcd2. 
Equating  this  to  the  external  moment  (125),  and  solving  for  d 


(c  +  v)/kc  (126) 

Ordinarily  this  depth  is  less  than  a  certain  minimum  necessary 
for  good  construction  and  a  minimum  depth  of  from  124  to  18 
inches  is  usually  specified  to  make  the  working  conditions  fa- 
vorable for  good  concrete  work  (see  later  sections) . 

The  shear  (see  equation  123),  is  found  to  be 

V  =  \  J  X  m(l  +  cx)g  =  *  mh(c  +  v)g 

From  (105)  the  necessary  value  of  d  is 

d  ,  m<Mc±j)  (127) 

Qjs 

Since  the  beams  are  comparatively  short  (the  counterforts  are 
generally  spaced  about  8  to  10  feet  apart)  it  is  quite  likely  that 
the  unit  adhesion  stress  will  be  high,  and  may,  in  fact,  control 
the  thickness  of  the  concrete  and  the  spacing  of  the  reinforce- 
ment. 

The  use  of  the  preceding  formulas,  and  the  relative  value  of 
the  several  stresses  and  their  effect  upon  the  dimensions  of  the 
member  are  illustrated  in  some  problems  at  the  end  of  the  chapter. 

Footing. — The  loading  upon  the  base  slab  is  the  net  difference 
between  the  downward  weight  of  the  retained  fill  and  the  upward 


REINFORCED  CONCRETE  WALLS  99 

soil  pressure.  (In  this  work  the  weight  of  the  slab  itself  is  neg- 
lected, since  its  downward  weight  is  practically  reflected  in  the 
upward  soil  pressure  intensity  caused  by  this  weight.)  The  load 
distribution  upon  the  slab  is  quite  problematical,  and  the  net 
difference  as  stated  above  does  not  exactly  give  the  actual  loads. 
The  distribution  of  soil  pressures  is  of  course  conditioned  upon 
the  deflection  of  the  base  slab,1  so  that  at  those  portions,  where 
there  is  a  maximum  stiffness  of  base  there  will  be  less  pressure 
(other  things  being  equal)  .  Accordingly  for  the  counterfort  walls  , 
the  maximum  deflection  of  the  base  slab  will  occur  midway  be- 
tween the  counterforts  and  toward  the  heel  and  the  minimum  at 
the  counterforts  and  toward  the  junction  of  the  arm  and  footing 
slabs.  These  niceties  of  pressure  distribution  will  not  enter  into 
the  following  treatment  of  the  design  of  the  base  slab  but  they 
should  be  borne  in  mind,  and  it  is  permissible  to  let  the  true 
state  of  affairs  color,  more  or  less,  the  computations  involved  in 
the  design  of  this  slab.  Essentially,  however,  the  following 
analysis,  gives  a  simple  method  of  design,  with  probably  a  stronger 
section  of  base  than  is  actually  required,  but  not  enough  stronger 
to  justify  a  highly  refined  analysis.  It  may  again  be  emphasized, 
that  a  little  excess  section  may  be  sacrificed  to  simplicity  of 
analysis. 

So  long  as  there  is  not  a  uniform  distribution  of  soil  pressure, 
the  minimum  upward  pressure  occurs  at  the  heel.  Since  the 
downward  load  is,  to  all  intents,  uniformly  distributed,  the  maxi- 
mum net  intensity  of  load  occurs  at  the  heel.  Again,  the 
maximum  soil  pressure  occurs  at  the  toe,  and  since  its  intensity 
will  be  larger  than  the  downward  intensity  of  pressure,  there  will 
likely  be  a  net  difference  of  pressure  upon  the  base  of  consider- 
able magnitude  and  directed  in  an  opposite  direction  to  the  net 
pressure  at  the  heel.  This  may  be  brought  out  algebraically  as 
follows  (see  Fig.  46)  : 

The  unit  downward  load  is         gh(l  +  c) 

where  the  variables  have  the  usual  meaning  as  defined  in  the 
preceding  pages.  The  soil  pressure  intensity,  Sx,  at  a  point  x 
from  the  heel  is,  from  (39)  after  makng  the  proper  substitutiens, 


Sx  =  2gh(l  +  c)(l  -  i)    $e  -  1  +  3(1  -  2e)  ~        (128) 

JA   discussion  of  this  point  is  given  in  CAIN  "Earth  Pressures,  Walls 
and  Bins,"  p.  157. 


100  RETAINING  WALLS 

making  the  net  downward  load  at  the  point  x,  Px 

Px  =  gh(l  +  c)Jx  (129) 

where 

Jx  =  1  -  2(1  -  i)Ze  -  1  +  3(1  -  2e)  (130) 


The  maximum  net  downward  pressure,  at  the  heel,  PI,  is,  with 
x  =  0 

P1  =  gh  0  +  c)  Ji  (131) 

where 

Ji  =  1  -  2(1  -  i)(3e  -  1)  (132) 

and  the  maximum  upward  net  pressure,  at  the  toe,  P2,  with  x  =  w 

P2  =  gh  (1  +  c)  J2  (133) 

with 

J2  =  1  -  2(1  -  i)(2  -  3e)  (134) 

When  the  point  of  application  of  the  resultant  falls  within 
the  outer  third  of  the  base,  the  soil  intensity  at  the  end  of  the 
heel  is  zero  and 

Pi  =  gh(l  +  c)  (135) 

The  above  equations  determine  the  loads  to  be  used  in  de- 
signing the  longitudinal  strips  of  the  base  slab  and  with  ra  the 
distance  between  the  counterforts,  the  moment  is 

M  =  Pra2/12  (136) 

where  the  proper  value  of  P  from  the  preceding  equations  is  to 
be  used.     The  shear  is  P/2. 

Similarly  to  the  design  of  the  face  slab,  the  required  depth  of 
the  slab,  due  to  the  bending  moment  is 


d  -  mVJxgh(l  +  c)7l2  (137) 

A  theoretical  comparison,  based  upon  the  bending  moment 
requirements  only,  may  be  had  between  the  depths  of  the  base  and 
of  the  arm  slabs.  The  depth  of  the  face  slab  is  governed  by  the 
thickness  required  at  the  base  of  the  arm;  that  of  the  base  slab 
by  the  thickness  required  by  the  maximum  value  of  J.  When  the 
resultant  falls  at  the  outer  third  point,  or  within  the  outer  third 
the  value  of  Jxisl.  Denoting  the  respective  required  thicknesses 
of  face  and  base  slabs  by  dv  and  db  respectively,  comparing,  equa- 
tions (126)  and  (137),  after  placing  v  =  1,  there  is 


dv  =  4A/OA7*)  (138) 


REINFORCED  CONCRETE, 


101 


and  with  Jx  =  1,  this  relation  becomes 

dv  =  0.584  (139) 

This  relation,  however,  is  more  of  academic  than  practical 
interest,  since  it  will  be  found  that  the  thicknesses  of  these  slabs 
are  controlled  by  factors  other  than  the  bending  moments. 
Later  on  this  relation  will  serve  a  fairly  useful  purpose  in  obtain- 
ing relative  economy  of  the  several  wall  types,  for  which  purpose 
it  is  of  some  practical  application. 

Counterfort.  —  The  counterfort  is  designed  as  a  simple  canti- 
lever beam,  with  effective  depth  e  as  shown  in  Fig.  46.  For  the 
reasons  given  on  the  preceding  pages  no  other  refinement  is  de- 
sirable in  treating  this  member.  To  anchor  the  slabs  to  the  coun- 
terfort, rods  are  placed  as  shown  in  Fig.  51  of  a  section  sufficient 
to  hold  the  stresses  induced  by  the  loadings.  For  the  face  slab 
the  necessary  rod  area  to  hold  a  strip  of  face  bounded  by  the 
two  horizontal  lines  x\  and  x*  from  the  top  of  the  wall,  with  m 
the  distance  between  the  counterforts,  and  taking  the  earth  pres- 
sure coefficient  /  as  J£  is  (see  Fig.  46). 

*).  (140) 


f8  is  the  permissible  unit  steel  stress,  and  g  the  weight  of  the  earth 
per  cubic  foot.  Using  a  value  of  16,000  pounds  per  square  inch 
for  fs  and  100  pounds  per  cubic  foot  for  g,  this  last  equation  takes 
the  form 

tyy\ 

As  =  ~(     +  2h'  +  x,)(xz  -  Xl)  (141) 


To  anchor  the  base  slab  to  the  counterfort  it  is  noticed  (see  Fig. 
46)  that  beyond  the  point  A  the  slab  and  the  counterfort  are  in 
compression.  It  is  therefore  necessary  to  provide  anchorage  for 
the  portion  of  the  base  between  A  and  B,  only.  The  point  A  is 
located  as  follows:  The  soil  intensity  at  A  is  found  from  (128). 
At  the  point  A  this  intensity  is  equal  to  the  downward  intensity 
gh  (1  +  c)  .  Forming  this  equality,  and  solving  for  x 

1  -  2(1  -  i)(36  -  1) 

6(1  -  i)(l  -  2e) 
or 

x  =  Dw  (142) 


102 


.      RETAINING  WALLS 


TABLE  23.— VALUES  OF  "D' 


To  facilitate  the  computation  of  D,  Table  23  has  been  pre- 
pared covering  a  range  of  values  of  i  and  e.  The  total  rod  area 
necessary  to  hold  the  portion  of  the  slab  A B  to  the  counterfort 

is  then  that  area  required  to 
hold  the  net  difference  in  the 
upward  and  downward  load- 
ings between  these  two  points. 
Two  conditions  exist  (see  Fig. 
47) :  when  the  point  of  appli- 
cation of  the  resultant  force 
lies  within  the  outer  third, 
and  when  it  lies  without  the 
outer  third.  For  the  former 
case,  the  point  of  zero  soil 

intensity  has  been  given  by  equation  (41)  of  Chapter  2,  and  the 
net  difference  in  loading  is 


•\ 

0 

.1 

.2 

.3 

.4 

.5 

0 

.50 

.52 

.54 

.57 

.61 

.67 

.1 

.50 

.52 

.55 

.59 

.63 

.71 

.2 

.50 

.53 

.57 

.62 

.68 

.78 

.3 

.50 

.55 

.60 

.68 

.77 

.92 

.33 

.50 

.56 

.62 

.72 

.83 

1.00 

.4 

.50 

.59 

.71 

.86 

.95 

1.00 

3  1  -  2e 


T  =  mghw  (1  +  c) 

z 

which  may  be  written,  simply 

T  =  mghw(l  +  c)E, 
where  E  represents  the  fraction  in  the  above  equation. 


(143) 


(144) 


.DW-- 


, Dw    - 


FIG.  4 


Again,  when  the  point  of  application  of  the  resultant  pressure 
is  without  the  outer  third,  i.e.,  when  the  soil  distribution  is  a 
trapezoidal  one,  the  value  of  T  may  be  given  by 

T  =  mghw(l  +  c)E' 
where 

E'  =  D{1  -(1  -  i)  [2(3c  -  1)  +  3(1  -  2e)D]}  (145) 

Table  No.  24  gives  the  values  of  E  for  a  range  of  values  of 
i  and  e. 


REINFORCED  CONCRETE  WALLS 


103 


The  application  of  the  above  expressions  to  specific  problems 
is  given  at  the  end  of  the  chapter. 

The  required  rod  area  to  hold  the  load  T  is 


As  =  T/f. 


(146) 


TABLE  24.— VALUES  OP  "E' 


0 

.1 

.2 

.3 

.4 

.5 

.42 

.43 

.44 

.45 

.47 

.50 

.40 

.41 

.42 

.44 

.46 

.50 

.36 

.38 

.40 

.42 

.45 

.50 

.29 

.31 

.34 

.38 

.43 

.50 

.25 

.28 

.31 

.36 

.42 

.50 

.15 

.19 

.24 

.31 

.40 

.50 

where /„ is  the  permissible  unit  steel  stress. 

The  preceding  analysis,  involving  as  it  does  a  series  of  mathe- 
matical expressions,  is  not  to  be  taken  as  interpreting  with 
exactness  the  stress  system  in  the 
counterfort  wall.  The  difficulty  of 
attaining  such  exact  statement 
has  been  pointed  out  above.  The 
work  as  given  is  to  be  used  as  a 
logical  step-by-step  process  of 
taking  care  in  as  simple  a  way  as 
possible  the  stresses  that  are  in- 
dicated by  a  general  study  of  the 
wall.  The  equations  together  with 
the  tables  based  upon  them  are 

readily  applied  to  numerical  problems  (as  given  at  the  end  of 
this  chapter)  and  cover  in  sufficient  detail  the  necessary  work 
in  determining  the  wall  dimensions  and  the  size  and  distribution 
of  the  rod  system. 

Rod  System, — The  anchorage  of  the  rod  system  into  the  wall 
members  forms  the  vital  part  of  the  design  of  the  counterforted 
wall.  While  it  may  seem  a  simple  feat  to  anchor  such  rods  to  the 
face  slab  (note  that,  in  what  follows,  particular  stress  is  laid 
upon  the  face  slab ;  the  thickness  of  the  base  slab  is  such  that  ample 
room  is  had  for  anchorage  of  the  tie-rods  by  simple  extension 
of  their  length  and  no  further  treatment  is  thus  required)  by 
threading  their  ends  and  bolting  through  steel  plates  or  washers 
or  even  to  assembled  steel  sections;  or  by  bending  around  rods 
at  right  angles  to  the  anchoring  rods,  such  details  involve  costly 
field  work,  the  use  of  an  expensive  class  of  labor  and  slow  up  to  a 
considerable  extent  the  progress  of  the  work.  Simple  details  are 
essential.  In  a  problem,  discussed  in  some  detail  at  the  end 
of  the  current  chapter,  a  detail  is  given  showing  such  anchor 
rods  bent  into  Us  of  a  radius  large  enough  to  prevent  crushing 
of  the  concrete  and  lying  in  a  vertical  plane.  Rods  of  small 
thickness  are  usually  used  because  of  the  greater  total  surface 
presented  for  adhesion. 


104  RETAINING  WALLS 

Problems 

1.  A  wall,  of  height  25  feet,  retains  an  ordinaiy  railroad  fill  subject  to  a 
surcharge  of  600  pounds  pei  square  foot.  It  is  placed  along  the  easement 
line,  beyond  which  no  encroachment  is  permissible.  The  soil  is  a  sandy 
loam  on  which  four  tons  per  squaie  foot  is  allowable  (see  Table  7).  A 
design  as  a  "L"  shaped  cantilever,  and  as  a  counterforted  reinforced  con- 
crete wall  is  desired. 

With  the  above  data  c  =  6/25  or  0.24;  i  =  0.  From  (93),  page  83,  with 
H  =  31  and  i  =  0,  the  location  of  the  resultant  is 

e  =  2/3  -  40/93  =  0.24 

With  this  value  of  e  and  with  i  =  0,  the  factoi  of  safety  against  overturn- 
ing is  2  (Table  17),  a  satisfactory  one  according  to  Hool,  but  less  than  the 
2.5  suggested  by  Cain.  See  page  57,  Adopting  this  value  of  e,  from  Table 
18  the  required  value  of  k  is  0.57  and  accordingly  the  base  will  be  made 
14  ft.  wide. 

From  Table  21  the  shear  and  the  bending  moment  require  about  the 
same  depth.  Using  the  shear  equation  (113) 

d  =  0.0033  X  252  X  1.48  =  3.05 

and  the  thickness  of  the  base  will  be  taken  as  three  feet.  At  a  point  half- 
way up  the  wall  for  which  c  =  ^{2-5  =  0.48,  the  moment  determines  the 
depth  at  this  point,  as  can  be  seen  from  Table  21,  and  from  (112) 

d  =  0.0185  X  44.2  X  \f(l  +  1.44)  =  1.28 

For  the  sake  of  simplicity  of  forms,  bracing  and  rods,  the  wall  will  be 
given  an  unbroken  batter  from  the  coping  to  the  base,  with  the  top  width 
a  minimum  practical  width  of  one  foot. 

At  the  midpoint  just  investigated,  the  thickness  will  then  be  two  feet,  in 
place  of  the  required  1.28  feet.  In  the  final  design  of  the  wall,  the  rod  sec- 
tion will  be  diminished  to  allow  for  the  decreased  moment. 

Footing. — From  (111,118)  the  required  depth  will  be  \//  times  the  depth 
necessary  for  the  arm  (since  the  arm  depth  here  is  that  practically  demanded 
by  the  moment).  From  Table  22,  since  i  =  0,  7  =  1,  and  the  depth  will 
be  the  same  as  that  required  of  the  arm  at  its  base,  namely  3'  0"  This 
thickness  will  be  maintained  to  the  end  of  the  heel. 

Enough  data  has  now  been  gathered  to  prepare  an  exact  and  final  design. 
From  table  3,  for  c  =  0.27,  B  =  0.40,  whence  Bh  =  0.40  X  22  =  8.8. 
Note  here  that  the  exact  length  of  the  arm  is  now  considered,  proper  allow- 
ance having  been  made  for  the  thickness  of  the  footing.  The  batter  of  the 
back  is  two  feet  in  twenty-two  feet,  or  b  =  tan~x  (Ki)  =5°  40'  =  6°. 
From  Table  1,  J  =  0.345  and  6  =  9°.  The  value  of  the  thrust  T  is,  from 
(14),  16  kips,  and  is  inclined  at  an  angle  of  15°  (d  +  6)  to  the  horizontal. 
The  weight  G  of  the  supei  imposed  earth  on  the  footing  is  22  +  6  =  28  X 
11  X  0.1  =  30.8  kips.  The  weights  of  the  footing  and  of  the  rectangular 
and  triangular  portions  of  the  arm  are  respectively,  6.3,  2.2  and  2.2  kips 
(see  Fig.  48).  Graphically,  the  resultant  is  found  to  intersect  the  base  3.5 
feet  from  the  toe  or  exactly  M  of  the  distance  from  the  toe,  checking  the 


REINFORCED  CONCRETE  WALLS 


105 


first  assumption.     The  horizontal  and  vertical  components  of  the  resultant 
found  graphically  are  respectively  15.8  and  46.5  kips.     With  the  latter 

93 

value  and  using  equation  (39)  Si  =  77  (2  -  0.75)  =  8.3  kips,  a  permissible 

14 

variation  from  the  4  tons  or  8  kips  assumed. 


'-53 


FIG.  48. 

Resistance  to  Sliding. — The  coefficient  of  friction  between  sandy  loam 
and  concrete  is  about  0.5  (an  average  between  sand  and  gravel),  see  Table  6. 
The  sliding  resistance  is  then  0.5  X  46.5  =  23.2  kips.  The  horizontal  com- 
ponent is  15.8  kips,  giving  a  factor  of  safety  against  sliding  of  23.2/15.8  = 
1.5  which  is  ample. 

Design  of  the  Vertical  Arm. — The  actual  loading  on  the  arm  extends  to 
the  top  of  the  footing  and  for  the  arm,  h  is  22'  and  c  —  0.27.  From  (110) 
with  J  =  }<j,  M  =  106.5  kip  ft.  and  the  shear  is  12.4  kips.  Taking,  as 
before,  the  steel  ratio  for  balanced  reinforcement,  or  p  =  0.75  per  cent., 
O.Sfc;  =  0.16  and  kc  =  105.  With  fc  =  12",  the  required  depth  d  in  inches 

=  106,500  X  12 

12  X  105 
From  (113)  the  depth  required  on   account  of  shear  is1 


whence  d  =  32' 


12,400 


=  28" 


0.89  X  12  X  40 

(From  Table  20  with  c  =  0.27,  the  shearing  stress  governs,  when  h  is  greater 
than  27'.) 


106  RETAINING  WALLS 

The  steel  area  required  at  B  is  0.0075  X  32  X  12  =  2.88  square  inches. 
This  is  a  rather  heavy  reinforcement,  necessitating  great  expense  in  handling 
and  placing  bars.  If  a  thicker  wall  is  assumed,  e.g.,  d  =  40",  then,  from 

the  properties  of  the  section  ks  =  10^5^^212  =66.     pj  =  66/16,000   = 

0.004  and  the  required  percentage  of  steel  is  0.4  per  cent.  The  steel  area 
is,  then,  0.004  X  40  X  12  =  1.92  square  inches  and  1  inch  square  bars  on 
6"  centers  will  give  the  necessary  area.  The  unit  adhesion  is 

12,400 
.89  X  40  X  8 

The  permissible  stress  is  80  pounds  per  square  inch.  To  determine  at  what 
point  it  is  possible  to  stop  one  half  of  these  rods,  i.e.  to  space  the  rods  12" 
apart,  note  that  the  external  moment  is  given  by  the  expression  (110)  or 

M  =  66.7(1  +  3c)z3 

Since  the  coping  width  is  taken  as  12",  the  effective  thickness  at  any  point 
of  the  wall  x  is 

d  =  9"  +   J2  (4°  ~  9)  =  9"  +  l'5x' 

The  resisting  moment  is  given  by  M  =  ksbdz,  where  ks  =  fapj.  For  small 
values  of  p,  pj  may  be  taken  equal  to  p,  and  with/,  =  16,000  pounds  per 
square  inch,  and  noting  that  since  the  area  of  steel  is  to  be  one  square  inch, 
p  =  l/bd,  the  resisting  moment  becomes 

M  =  16,000d  =  16,000  (9  +  1.5:e) 

Equating  the  resisting  moment  to  the  external  bending  moment  and  replac- 
ing c  by  its  value  6/z,  there  results  a  cubic  in  x 

x*  -f  18z2  -  360z  -  2160=  0 

which  is  satisfied  by  x  =  15.  Accordingly,  at  a  point  15'  below  the  top  of 
the  arm  the  rods  will  be  spaced  12"  apart.  Since  a  further  reduction  in 
the  spacing  would  place  the  rods  24"  apart,  which  is  not  good  practice,  the 
12"  spacing  will  be  continued  to  the  top  of  the  arm. 

Footing. — To  analyze  the  footing  stresses,  a  moment  diagram  has  been 
drawn  in  Fig.  48.  Note  that  the  moment  at  B  is  very  nearly  equal  to  the 
arm  moment  at  B,  affording  another  check  upon  the  approximate  method. 
With  Mb  ='  110  foot  kips,  and  for  balanced  reinforcement,  the  required 
depth  is  34".  The  necessary  amount  of  steel  is  0.0075  X  34  X  12  =  2.73. 
This  again  demands  too  heavy  a  reinforcement  for  efficient  handling,  and  a 

110  000  X  12 
thicker  concrete  will  be  assumed.     With  d  =  40,  k8  =  — 12  X  402     =  69' 

and  pj  =  69/16,000  =  0.0043.  The  steel  area  is  then  2  square  inches  and 
one  inch  bars  spaced  6"  apart  will  be  used.  To  determine,  again,  at  what 
point  it  will  be  possible  to  reduce  the  steel  section  to  one-inch  bars  at  12" 
spacing,  the  resisting  moment  of  such  a  steel  section,  since  the  thickness  of 
the  base  is  kept  constant,  is  found  to  be,  with  p  =  I/ (12  X  40)  =  0.0021. 
M  =  0.0021  X  16,000  X  12  X  402  =  635  inch  kips  or  53  foot  kips. 
Plotting  this  value  upon  the  moment  diagram  of  the  footing,  it  is  found  that 
at  a  point  6  feet  from  the  heel  it  is  possible  to  reduce  the  rod  section  to  one 


REINFORCED  CONCRETE  W  ALLS  107 

inch  bars  12"  apart.  For  the  reasons  outlined  above,  there  will  be  no  further 
increase  in  this  spacing. 

To  develop  the  adhesion  in  the  vertical  and  horizontal  rods,  which  must 
be  carried  out  50  thicknesses  or  4'  beyond  the  point  of  maximum  moment, 
it  is  necessary  to  place  a  6"  projection  at  the  toe  and  into  the  footing  as 
shown  in  Fig.  50. 

The  spacing  of  the  secondary  rod  system  for  shrinkage,  settlement  and 
temperature  will  be  discussed  in  a  later  chapter. 

Counterforted  Wall.  —  Adopting  the  economical1  spacing  of  ten  feet  for 
the  counterforts;  from  (126),  with  v  =  1,  kc  =  16,000,  h  =  25, 

d  =  0.7  feet. 

It  is  impractical  to  pour  concrete  in  a  wall  this  thick  for  the  height  as  given 
and  a  minimum  thickness  of  12"  will  be  adopted. 

From  (139)  tne  required  thickness  of  the  footing  slab  is  \/3  times  that 
required  of  the  vertical  slab.  It  will  be  seen  later  that  this  thickness  will 
be  controlled  by  a  thickness  necessary  to  get  a  practical  spacing  of  rods  for 
adhesion.  The  dimensions  of  the  separate  members  as  now  found  are  less 
than  those  of  the  cantilevered  wall,  and  since  that  wall  as  finally  designed 
agreed  with  the  approximate  dimensions  it  is  clear  that  the  counterforted 
wall,  will  likewise  agree  and  it  will  not  be  necessary  to  recheck  the  outline 
dimensions  of  the  section. 

In  selecting  rod  systems,  both  spacings  and  sizes,  and  wall  thicknesses, 
it  must  be  borne  in  mind  that  there  must  be  sufficient  working  space  to 
pour  the  concrete;  that  small  sizes  of  rods  are  relatively  more  expensive  than 
the  larger  sizes;  that  many  variations  in  both  length  and  spacing  tend  to 
cause  confusion  in  construction.  This  limitation  of  the  economical  section 
on  paper  by  field  conditions,  is  discussed  more  in  detail  in  the  following 
chapter. 

The  moment  at  the  base  of  the  vertical  slab  (here  h  =  23.5  feet),  with 

100  X  2Q  5 
c  =  .265,  and  P  =       -—-'       =  0.98  kips,  is  from  (125)  8.3  kip  feet.     As 

before  the  depth  for  this  moment,  with  balanced  reinforcement  is  0.73  feet, 
but,  for  reasons,  outlined  above  the  thickness  will  be  taken  as  12".  With  a 
wall  of  this  thickness  the  utmost  care  must  be  exercised  in  pouring  concrete 
into  it.  See  Chapter  VIII  for  the  precautions  to  be  used  to  insure  a  well 
mixed  and  rammed  concrete. 

With  a  depth  to  steel  of  10",  kc  =  8300  X  12/12  X  100  =  83  and  p  = 
0.004,  which  gives  a  required  area  of  0.48  square  inches,  which  %"  rods  on 
12"  centers  satisfies. 

The  total  shear  is  980  X  5  =  4900  pounds,  and  the  unit  shear  from  (105)  is 

4900 
0.8<O<  10  X  12  =        P°unds  Per  square  inch 

which  is  so  slightly  in  excess  of  the  permissible  stress  of  40  pounds,  that  the 
section  will  be  maintained  as  assumed.  The  area  required  for  adhesion  is 
from  (106) 

4900 

mches 


See  problem,  Chapter  IV,  p.  150. 


108  RETAINING  WALLS 

The  adhesion  stress  thus  governs  the  spacing  of  the  rods  and  %"  rods  spaced 
5"  apart  will  give  the  required  periphery  of  section. 

At  h  =  15,  the  moment  is  5.8  kip  feet  and  the  shear  is  700  X  5  =  3.5 
kips.  The  area  required  for  the  bending  moment  is  accordingly  0. 18  square 
inches,  while  that  required  for  adhesion  is  found  to  be  5  inches. 

At  h  =  10  feet,  the  periphery  required  for  adhesion  is  3.8  and  at  h  =  5 
feet,  the  required  periphery  for  adhesion  is  2.6  inches. 

It  is  seen  that  the  adhesion  stress  will  determine  the  spacing  of  the  rods 
throughout  the  arm.  At  h  =  15,  since  5  inches  are  required  for  adhesion 
the  spacing  at  the  base  will  be  maintained  beyond  this  point.  At  h  =  10 
feet,  since  r  =  3.8  the  rods  may  be  spaced  on  10"  centers.  At  h  =  5'  the 
value  of  r  required  will  not  permit  a  further  reduction  in  the  spacing  of  the 
rods.  There  will  thus  be  %"  square  rods  spaced  on  5"  centers  from 
the  base  to  h  =  10  feet  and  ten  inch  spacing  from  there  to  the  top  of  the 
wall.  To  take  care  of  the  equal  but  negative  moment  at  the  counterfort, 
with  the  corresponding  adhesion  stresses,  the  same  spacing  will  be  main- 
tained on  the  inner  face  of  the  vertical  slab.  Since  the  rods  must  be 
carried  beyond  the  point  of  zero  moment  (approximately  the  quarter  point) 
the  rods  on  the  inner  face  will  be  made  five  feet  long  centered  at  the 
counterforts. 

Footing. — The  net  weight  on  the  footing  excluding  the  excess  weight  of 
the  masonry  over  the  earth,  is  3100  pounds.  As  before  a  depth  to  satisfy 
the  benoling  moment,  is  from  (138)  18".  For  adhesion 

3100  X  5 
.88  X  16  X  80 

which  cannot  be  readily  and  practically  provided.  Conversely  since  it  is 
desirable  to  use  a  rod  not  exceeding  the  section  of  %"  rod  whose  minimum 
spacing  is  5"  on  center,  d  is  found 

3100  X  5 
0.88X7.2X80 

and  the  total  depth  of  the  footing  slab  is  thus  30  +  2"  =  32". 

The  point  where  the  upward  and  downward  intensities  balance  each  other 
is,  from  (142)  and  Table  No.  23  with  i  =  0  and  e  =  Y±,  at  the  midpoint  or 
seven  feet  from  the  end  of  the  heel.  To  avoid  many  changes  in  the  spacing 
of  the  rods,  the  %-inch  square  rods  will  be  spaced  on  5-inch  centers  to  a 
point  3.5  feet  from  the  heel  and  thence,  to  the  midpoint  on  10  inch  spacing. 

For  the  portion  between  the  midpoint  and  the  vertical  arm  it  is  reasonable 
to  assume  that  the  slab  is  supported  on  three  edges — the  counterfort  edges 
and  the  vertical  arm — and  that  such  support  is  uniformly  distributed  along 
such  edges.1  From  (134)  J2  =  1  -  2(2  -  %)  =  -1.5.  P2  =  1.5  X  100 
X  28  =  4200  Ib.  The  total  net  load  between  the  counterforts  reacting 
upward  upon  the  slab  is  then,  since  the  intensity  is  zero  at  the  midpoint, 
4.2  X  7  X  9/2  =  132  kips.  The  total  length  of  supporting  edge  is  2  X  7  + 

1  For  an  interesting  discussion  of  this  modification  of  plate  theory  it  may 
be  well  to  consult  Prof.  Eddy's  brilliant  little  book  on  the  "Theory  of  Rec- 
tangular Plates." 


REINFORCED  CONCRETE  WALLS  109 

9  =  23  feet  and  the  shear  per  linear  foot  is  5.7  kips.  For  a  30"  slab  the  unit 
shear  is  then  — —  — —  =  18  Ib.  considerably  below  the  allowable 

.89  X  12   X  oO 

5700 

and  the  periphery  of  rod  required  for  adhesion  is  -— — — -  =2.7 

.89  X  80  X  oO 

square  inches.  It  is  then  sufficient  to  carry  the  %"  rods  on  10"  spacing 
to  the  toe  of  the  base.  The  rod  spacing  will  be  duplicated  on  the  opposite 
face  to  take  care  of  the  negative  moment  and  reversed  stresses.  Thus 
from  the  midpoint  out  to  the  heel  the  rods  on  the  lower  face  will  be  carried 
full  length  and  those  on  the  upper  face  five  feet  beyond  the  counterfort. 
From  the  midpoint  to  the  heel  the  rods  on  the  lower  face  will  be  carried 
for  the  full  length  and  those  on  the  upper  face  will  be  extended  five  feet  on 
either  side  of  the  counterfort. 

Counterfort. — Designed  as  a  cantilever  beam,  the  moment  at  the  base 
is  then  TBh,  with  the  thrust  taken  for  a  length  m  of  the  wall.  T  =  10  X 
13  =  130  kips.  With  c  =  %2  =  0.27;  B  =  0.39  and  h  =  22, 

M  =  130  X  0.39  X  22  =  1,140  kip  feet 

The  depth  e  of  the  cantilever  is  14'.  Assume,  tentatively,  its  thickness 
as  1.0  feet. 

kc  =  41,  making  p  =  0.001  and  the  required  area  of  steel  in  square  inches 
is  2  square  inches.  Therefore  two  inch  square  bars  are  ample  to  take  care 
of  the  moment  in  this  counterfort.  Investigating  the  unit  adhesion,  it  is 
found  that,  with  a  value  of  V  =  130,000  pounds,  the  unit  adhesion  is  110 
pounds.  If  two  \Y±'  bars  are  used,  the  unit  adhesion  is  found  to  be  85 
pounds  per  square  inch,  a  permissible  variation  from  the  allowable  80  pounds. 
To  anchor  these  rods  into  the  base  it  is  necessary  to  carry  them  fifty  thick- 
nesses or  about  five  feet  into  the  foundation.  For  this  reason  an  extension 
will  be  built  into  the  foundation  two  feet  below  the  slab  and  carried  six 
inches  on  either  side  of  the  counterfort.  The  radius  to  which  the  rods  must 
be  bent  in  going  into  the  base  slab  is  30  X  134  =  3'  0". 

To  anchor  the  face  slab  to  the  counterfort,  since  the  thickness  of  the  face 
slab  does  not  permit  a  straight  extension  of  the  rods  into  it,  it  will  be  neces- 
sary to  adopt  the  expedient  of  bending  the  rods  into  a  U,  with  the  radius 
of  the  curve  30*. 

From  (141)  for  the  top  five  feet  of  the  wall 

Aa  =  ^°0  (5  +  12)  X  5  =  0.89 

Therefore  two  M"  Us  give  sufficient  bond  for  this  length.  The  bars  must 
be  bent  to  a  radius  of  15".  For  the  next  five  feet  the  required  amount  of 
steel  is  1.4  and  two  %"  rods  bent  to  a  U  with  radius  of  18"  pro  vide  the  requisite 
bond.  For  the  five  feet  below  this  section  Aa  =  1.9,  and  three  %"  Us  as 
shown  in  Fig.  51  satisfy  the  requirements  of  this  portion.  The  remaining 
space  from  15'  to  22'  is  divided  into  two  parts,  the  area  of  the  first  part  is 
found  to  be  1.6,  of  the  lower  part  1.9.  Therefore  three  %"  Us  as  previously 
detailed  .will  provide  the  remaining  bond  rods. 

To  gee  the  necessary  rod  area  to  anchor  the  heel  portion  of  footing  to  the 
counterfort  (the  portion  from  the  midpoint  to  the  heel)  from  (144)  and 


110 


RETAINING  WALLS 


Table  24  with  E  =  0.33  the  total  load  to  be  held  by  these  rods  is  10  X  100  X 
28  X  0.33  X  14  =  129  kips.  The  steel  area  is  then,  from  (146)  129/16  = 
8. 1  square  inches.  Using  ^  square  rods,  one  on  either  side  of  the  counter- 
fort 32  are  required.  With  6"  spacing  15  spaces  will  carry  the  rods  beyond 
the  midpoint.  The  depth  of  the  footing  is  ample  to  develop  these  rods  in 
adhesion  without  any  special  detail  and  they  will  be  carried  to  two  inches 
from  the  bottom,  of  the  footing.  Theoretically  they  need  be  carried  into 
the  counterfort  the  same  distance,  but  it  seems  better  practice  to  carry 
the  rods  for  the  full  height  of  the  portion  of  the  counterfort  affected  (see 
Fig.  51). 

2.  Modify  the  preceding  problem  to  carry  a  railroad  track  system  with 
wall  track  8  feet  away  from  the  face  of  the  wall  and  the  other  tracks  on  12.5 
foot  spacing.  Assume  that  all  tracks  but  the  wall  track  are  loaded;  then 
assume  no  tracks  loaded.  In  what  way  is  the  pressure  upon  the  footing 
affected,  and  do  any  of  the  stresses  exceed  those  for  the  case  of  all  tracks 
loaded  (the  former  case)  ? 


Dotted  Lines  show 
Force  Polygon  for 
fio  Surface  Loading, 


T-  8.0  Kips 


T=I0.6      /  /T=8.0 


FIG.  49. 


For  this  case,  see  Fig.  10,  the  surcharge  extends  to  14  feet  from  the  wall 
face.     As  above  6  =  6°  and  from  (32)  of  Chapter  I 


The  proper  value  of  a  to  use  in  determining  the  coefficient  K  is  from  (34) 
with  y  =  14/22  =  0.64 

tan  a  =  tan  6°  -  —5  0.64  =  -  0.119 


whence 


a  =  -7 


From  table  No.  13,  allowing  no  friction  upon  the  back  of  the  wall,  K  *=  0.285 
and  the  thrust  is  then  10.6  kips.  Fig.  49  shows  the  force  system  on  the 
wall  for  this  case. 


REINFORCED  CONCRETE  WALLS 


111 


For  the  second  condition,  no  loading  upon  the  surface,  the  thrust  becomes, 
with  K  =  0.33  and  c  =  0,  T  =  8  kips.  Fig.  49  shows  the  force  system 
for  this  case. 


FIG.  50. — Cantilever  wall. 


FIG.  51. — Counterfort  wall. 


*f 

^4$ 
u^  5-> 

^H.^ 

•T  1 

Front  El 
Vertical 

svation 
Slab 

Rear  Ele 


Vertical  Slab 


'at  ion 


Lower  Face 
Footing 

FIG.  52. — Rod  layout  counterfort  wall. 


Upper  Face 
Footing 


From  Fig.  49,  e  for  the  former  condition  is  0.28  and  R  =  37.5.  For  the 
second  condition  e  =0.35  and  R  =  37.  From  (38)  Si  for  the  former  is 
6750  pounds  per  square  foot  and  for  the  latter  is  5000  pounds  per  square 


112 


RETAINING  WALLS 


foot.     It  is  obvious  that  the  analysis  of  the  first  problem  will  require  no 
modification  of  stress  distribution  because  of  these  latter  conditions. 

Fig.  50  gives  the  detailed  layout  of  the  "L"  shaped  cantilever.     Fig. 
51  gives  the  rod  layout  of  the  counterfort  and  Fig.  52  of  the  vertical  and 
base  slabs.     In  neither  of  the  sketches  are  the  temperature  and  check  rods 
shown.     A  later  chapter  will  indicate  such  distributions. 
3.  A  "T"  shaped  cantilever  wall  is  to  be  built,  retain- 
ing an  embankment  as  shown  in  Fig.   53.     The  em- 
bankment is  subject  to  a  surcharge  live  load  of  750 
pounds    per    square   foot.       The    foundation    pressure 
must  not  exceed  5000  pounds  per  square  foot.      Deter- 
mine the  proper  wall  dimensions  and  details. 

For  the  condition  of  no  surcharge,  both  the  exact 
and  the  approximate  expressions  for  the  thrust,  as  given 
on    page    14    may  be  employed.      Exactly,  with  the 
angle  i  =  30°,  &  =  <f>'  =  0  and  0  =  30°,  L  =  I/cos2  <£  =  4/3;  u  -  sin  </>;  v  = 
—cos  <t>;  d  =  cot  <£;  m  =  1;  n  =  —cot2  <j>  =  -3;  c  =  0.5;  p  =  sin  <j>  =  K 
and/  =  —  3.     The  expression  for  the  thrust  is  then 


Fia.  53. 


=^  x|  X  [l.5  -  0.5  Vl.52  +  0. 


0.25 


=  10.7 

The  approximate  method,  which  since  c  = 
15),  gives  a  value 

T  =       (1  +  2c) 


0.5,  is  not  to  be  used  (see  page 


13.3. 


A  variation  from  the  true  value  too  excessive  to  permit  of  its  use. 

For  the  condition  of  a  live  load  surcharge, 
in  place  of  the  graphical  method  of  obtain- 
ing the  thrust,  the  compromise,  algebraic 
geometric  method  outlined  in  the  problem 
at  the  end  of  Chapter  1,  may  be  used.  The 
value  of  i  is  determined  graphically,  the 
line  forming  the  equivalent  triangles  as 
shown  in  Fig.  54.  With  aoe  making  an 
angle  of  35°,  the  triangles  afo  and  obe  are 
equivalent.  With  this  value  the  thrust  may 
be  determined  as  above.  From  Eq.  22  L  = 


FIG.  54. 


I/cos2 


u  =  sn 


—cos  <£;  n  =  —cot  35°  cot  $  =  —  2.43;  p  =  sin 


<f>  =  0.5;  m  =  1  and/  =  —2.43. 
100  X  400  _ 


L8752  +  0.761  X  2.43 


=  13.6 

Refer  to  Figs.  42  and  53  assuming,  as  the  condition  of  economy,  that  i  =e. 
In  addition,  assume  that  the  resultant  intersects  the  base  at  the  outer  third 
point,  i.e.  i  =  %.  Noting  that  g  =  100;  h  =  20  and  tan  35°  =  0.7  the 
weight  G  has  the  value 

M        -V  i  jj  0<l  ~  W™*ton  35° 
G  =  0(1  -  ^)wh  +  — 2 — 

=  0.67  w(2  +  0.023w)  (A) 


.REINFORCED  CONCRETE  WALLS  113 

Taking  moments  about  0,  and  noting  that  without  serious  error  the  point 
of  application  of  the  weight  may  be  taken  at  the  middle  of  the  base 

G(l  -  i)w/2  =  Th/3. 
Introducing  the  values  above,  this  equation  becomes 

181.5 
(1  -  i)w 
and  with  i  =  }i 

G  =  273/w  (£) 

Equating  (A)  and  (£),  there  results  a  cubic  in  w 

410  =  2w*  +  0.023w3 

which  is  satisfied  by  the  root,  w  =  13.5.  With  this  value  of  w,  G  =  20.2 
and  from  (39) 

Si  =  2G/w  =  40.4/13.5  =  3  kips. 

The  projection  of  the  toe  beyond  the  face  of  the  wall  is  4'  6".  Assume 
tentatively  that  the  thickness  of  the  base  and  of  the  vertical  ar  i  at  its  base 
is  two  feet.  The  thiust,  for  the  purposes  at  hand  may  be  assumed  to  vary 
as  the  square  of  h.  Since  the  effective  height  of  the  wall,  so  far  as  the  arm 
is  concerned  is  18  feet, 

102 

T  =  2™  X  13.6  =  11. 

and  its  point  of  application  is  one-third  of  h  or  6  feet  above  the  top  of  the 
footing.  The  bending  moment  is  then  11  X  6  =  66  and  with  k  =  16,000 
for  balanced  reinforcement,  the  required  depth  on  account  of  moment  is 

d  =  V(66/16)  =  2.03 

The  shear  is  11,000  pounds  and  the  depth  to  satisfy  this  amount  is 
d  =  11,000/5040  =  2.18 

The  thickness  of  the  vertical  arm  at  its  base  may  be  taken  as  2'  6".  The 
back  will  be  battered  to  a  top  thickness  of 'one  foot. 

Footing. — The  face  of  the  vertical  arm  is,  on  the  assumptions  previously 
made  at  the  third  point  or  4'  6"  from  the  end  of  the  toe.  The  moment  of 
the  heel  cantilever  is  then  taken  at  a  point  4'  6"  +  2'  6"  from  the  toe  or 
6'  6"  from  the  end  of  the  heel.  At  this  point,  since  Si  is  3000  pounds,  the 

6.5 

soil  intensity  is X  3  =  1.44. 

13.5 

Taking  the  approximate  value  of  G  as  20.2  and  again  assuming  that  it  is 
directed  over  the  center  of  the  heel  cantilever,  the  bending  moment  becomes 

i  4.4  v  fi  ^ 

20.2  X  3.25  -  -       *         X  2.2  =  55.4 
4 

The  shear  is 

20.2  -  1.44  X  6.5/2  =  15.5. 

Evidently  the  shear  will  control  the  depth  required  and 
d  =  15,500/5040  =  3.08 


114 


RETAINING  WALLS 


Whence  take  3'  as  the  required  thickness  of  base. 
It  is  now  possible  to  proceed  with  the  exact  design.     (See  Fig.  55.) 
thrust  is  found  from  equation  (22),  with  c  =  17.5/17  =  1.03  and 


The 


x|[2.03  -  2^2. 


2.03*  +  LOG  X  3 


r- 


8.9 


This  will  be  applied  at  a  point  17/3  or  5.65  feet  above  the  top  of  footing. 
The  weights  of  the  earth  has  been  divided  up  into  the  triangles  abc  =  d; 
ade  =  Gz  and  the  lectangle  dcfe  =  G2.  The  weight  of  the  masonry  has 
been  divided  into  the  triangle  G6  and  the  rectangles  G4  and  G6.  The  weights 
are: 

•Gi  =  9  X  4.75  X  100/2  =  2.14  kips. 

G2  =  6  X  17  X  100         =  10.2  kips. 


FIG.  55. 

Note  that  the  two  above  act  in  practically  the  same  vertical  line,  so  that 
the  two  may  be  added  and  treated  as  one  force 

Gi  +  G2  =  12.3 
<73  =  2  X  17  X  100/2  =  1.7 
G4  =  1  X  17  X  150  =  2.55 
G5  =  1  X  17  X  150  =  2.55 
G6  =  3  X  13.5  X  150  =  6.07 

With  the  forces  as  above  found  the  polygon  is  drawn  in  the  usual  manner, 
see  Fig.  55,  and  the  location  and  amount  of  the  resultant  pressure  is  found. 
The  actual  value  of  k  is  5.5/13.5  =  0.4  and  R  =  25.5. 


(2  -  0.12)  =  3.00  and  S2  = 


(1.2  -  1.0)  =  0.75 


.  . 

lo.O  lo.O 

Vertical  Arm.  —  The  moment  of  the  thrust  is  8.9  X  5.65  =  50.4  and  the 
depth  to  satisfy  this  moment  is 

d  =  V  (504716)  =1.78 

The  shear  is  8900  and  the  corresponding  depth  required  is 
d  =  8900/5040  =  1.77 


REINFORCED  CONCRETE  WALLS  115 

The  required  depths  are  thus  identical  and  the  total  thickness  of  slab  at 
the  base  of  the  arm  will  be  2'  0",  allowing  3"  for  a  protective  concrete  coat. 
Since,  for  balanced  reinforcement  the  steel  ratio  is  0.0075,  the  amount 
steel  required  is 

As  =  0.0075  X  21  X  12  =  1.89. 

Spacing  1  inch  square  bars  (deformed)  6"  apart  will  furnish  the  necessary 
section.  Assuming  that  there  is  a  triangular  distribution  of  pressure,1  the 
moment  diagram  is  shown  in  Fig.  55.  To  obtain  the  thrusts  for  the 
moment,  note  that  at  the  points  15',  10'  and  5'  from  the  top  of  the  wall  the 
corresponding  values  of  the  surcharge  ratio  are  1.17;  1.75  and  3.5.  The 
values  of  the  thrust  are  then 

Tu  =  IWJpl1  X  ||~2.17  -  »-\/  2.172  +  3  X  1.17*  1  '  =  6.9 

2]2  = 


Tlo  =        --—   X       2.75  -          2752  +  3  X      752       =  3.3 
T5    =  M<A*  x       4.5  -          4.52  +3X  3.52       =  0.9 


.5  -  2  V  4.52  +~3~X  3.52]  2  =  0. 


The  moments  are,  assuming  again  that  the  thrusts  are  }$  of  the  distance 
above  the  point  in  question, 

M15  =  6.9  X  5  =  34.5 
MM  =  3.3  X  3.3  =10.9 
M5  =  0.9  X  1.67  =  1.5 

At  some  intermediate  point  along  this  arm,  it  will  be  found  that  one  half 
of  the  rods  are  sufficient  to  carry  the  stress;  i.e.,  the  rods  from  this  point 
on  may  be  carried  on  12  inch  spacing.  As  before  the  width  of  the  wall  at 
the  coping  will  be  taken  as  12  inches.  With  a  spacing  of  12  inches  for  the 
one-inch  rods  at 

h  =  15;  d  =  19"     and     p  =  1/(19  X  12)  =  0.0044 

h  =  10'  d  =  16"  p  =  1/(16  X  12)  =  0.0052 

The  corresponding  values  of  pj  are  0.0042  and  0.0047,  and  the  resisting 
moments  are  then,  expressed  in  foot-pound  units, 

M15  =  144  X  0.0042  X  16,000  X  1.592  =  24.3 
Mio  =  144  X  0.0047  X  16,000  X  1.332  =  19.2 

Plotting  these  two  values  on  the  moment  diagram,  Fig.  55,  it  is  seen  that 
the  resisting  moment  of  one-inch  rods  on  twelve-inch  centers,  is  equal  to 
the  external  bending  moment  at  a  point  approximately  4.5  feet  above  the 
footing.  The  six-inch  spacing  will  then  be  stopped  at  a  point  5'  above  the 
top  of  the  base  slab.  As  previously  explained,  this  spacing  will  be  continued 
to  the  top  of  the  arm. 

1  While  this  is,  strictly  speaking,  incorrect,  since  the  thrust  is  not  a  linear 
function  of  h,  which  condition  is  the  necessary  one  that  there  be  a  triangular 
distribution  of  pressure,  the  ease  of  handling  the  problem  with  that  assump- 
tion counterbalances  the  slightly  excessive  pressures  thus  found. 


116  RETAINING  WALLS 

Footing.  —  The  force  acting  upon  the  base  slab  over  the  heel  is  G\  +  Gz 
or  12.3  kips.  The  weight  of  the  base  slab  (maintaining  the  thickness  first 
found)  is  6.5  X  3  X  150  =  2.9  kips.  The  total  downward  load  upon  the 

13X65 
heel  is  15.2  kips.     The  upward  soil  pressure  is   --  -  --  =  4.22  kips.     The 

moment  for  the  heel  is  thus 

15.2  X  3.25  -  4.22  X  4.33  =  31.2  kip  feet 

The  shear  is  15.2  —  4.2  =  11  kips.  The  required  depth,  for  shear  is  2.18, 
which  clearly,  is  greater  than  that  required  for  the  moment.  With  a  pro- 
tective concrete  over  the  rods  the  thickness  of  the  heel  slab  will  'be  taken 
as  30".  With  the  net  depth  (effective)  of  27",  ka  =  31.2/2.252  =  42.5  and 
pj  =  42.5/16,000  =  0.003.  The  steel  ratio  is  then  0.003  and  the  necessary 
section  of  rods  becomes  0.003  X  27  X  12  =  0.97  square  inches.  One-inch 
rods  spaced  twelve  inches  apart  will  provide  the  requisite  steel  area  and 
this  spacing  will  be  carried  out  to  the  end  of  the  heel. 
Toe.  —  At  the  toe  the  cantilever  moment  is 


and  the  shear  is 

(3  +2.3)4.5/2  =  11.9  kips 

As  before  the  shear  requirement  will  control  the  depth  of  the  section 
d  =  11.9/5040  =  2.W33 

The  same  thickness  of  both  heel  and  toe  will  be  used,  which  in'  view  of  the 
usual  manner  of  pouring  the  wall  is  practically  mandatory. 

_  26,800.X  12 

12  X  27^ 

and  pj  =  37/16,000  =0.0023.  The  steel  ratio  is  then  0.0023  and  the  area 
required  is  0.0023  X  27  X  12  =  0.83.  1-inch  bars  spaced  twelve  inches 
apart  will  provide  the  steel  reinforcement. 

Since  a  1"  bar  requires  four  feet  to  develop  its  tension  by  adhesion,  the 
heel  rods  will  be  carried  four  feet  beyond  the  rear  face  of  the  vertical  arm 
and  the  toe  rods  four  feet  beyond  the  front  face  of  the  vertical  arm.  For 
the  reinforcement  of  the  vertical  arm,  an  extension  1'  0"  wide  and  1'  6" 
deep  will  be  built  into  the  foundation  to  provide  the  required  length. 

Fig.  56  shows  the  complete  section  of  wall.  The  rods  necessary  for  shrink- 
age and  temperature  stresses  have  not  been  shown. 

4.  In  the  wall  of  problem  1,  it  will  be  necessary,  for  a  given  stretch  to 
provide  a  foot-walk  as  shown  in  Fig.  57.  Without  changing  the  outlines 
or  the  design  of  the  wall  proper,  design  the  bracket  to  carry  this  walk,  sub- 
ject to  a  live  load  of  100  pounds  per  square  foot. 

Assuming  that  the  concrete  bracket  will  be  6"  thick,  the  dead  load  will 
be  75  pounds  per  square  foot,  making  the  entire  load  upon  the  bracket  175 
pounds  per  square  foot.  For  balanced  reinforcement 

d  =  V  (790/1  6,000)  =0.22,  or  3"  thick. 
With  2"  protective  concrete  over  the  rods  the  total  thickness  of  slab  is  5". 


REINFORCED  CONCRETE  WALLS 


117 


The  required  steel  area  is  0.0075  X  3  X  12  =  0.27,  and  K  inch  square 
rods,  12"  apart  will  provide  the  required  steel  section.     The  unit  shear  is 

525 


X3  X  12 


16  pounds 


The  unit  adhesion,  with  r  =  2,  is 
525 


Q.  = 


.89  X  3  X  2 


98  pounds  per  square  inch. 


This  latter  value  is  excessive  and  the  depth  of  section  must  be  increased  at 
this  point.  If  at  the  cantilever  junction  between  the  wall  and  bracket  a 
fillet  is  placed  as  shown  in  Fig.  57,  the  unit  adhesion  at  the  point  D  is  %  of 
that  above  found  or  70  pounds  per  square  inch.  To  provide  the  necessary 
bond  the  V>"  rods  will  be  bent  as  shown  and  carried  into  the  vertical  arm. 


L- 


6'Cenfers 


4 


Reinforcement 
'of  Verh'calArm 


JJ 


I /"VfcHfe- 

~.  I2"Center£ 
13# 


FIG.  56. 


FIG.  57. 


5.  A  counterforted  wall,  resting  upon  a  rock  bottom,  is  to  take  a  surcharge 
of  500  pounds  per  square  foot.  The  easement  does  not  permit  a  toe  exten- 
sion. Determine  the  general  wall  outlines  from  the  approximate  formulas 
given  and  design  a  counterfort  made  up  of  a  steel  truss. 

With  i  =  0,  and  the  foundation  rock  e  may  be  taken  equal  to  ^, 
giving  a  value  of  k  from  Table  18  of  0.51.  The  width  of  base  is  thus  0.51  X 
50  =  25'  6".  From  Table  17  the  factor  of  safety  is  found  to  be  two.  As- 
sume that  the  counterforts  will  be  spaced  ten  feet  apart.  The  pressure  at 
the  base  of  the  vertical  slab  is  Jgh(l  +  c)  =  0.33  X  50  X  1.1  X  0.1  =  1.83 
kips  per  square  foot.  From  (126) 


1  100  X  50  X  1.1 
3     12  X  16,000 


=  1.0 


The  depth  for  shear  is 


d  = 


1.83  X  5 
5.04 


1.83 


118 


RETAINING  WALLS 


It  will  be  found,  later  that  the  thickness  of  the  face  slab  at  the  base  will 
be  controlled  by  the  necessary  dimensions  of  the  member  composing  the 
vertical  arm  of  the  truss.  The  thickness  of  the  base  slab  is  controlled  by 
the  depth  necessary  for  the  adhesion  stresses.  If  1"  square  bars,  spaced 
6"  apart  are  to  be  used,  then  the  depth  necessary  to  satisfy  the  limiting 
adhesion  stress  of  80  pounds  per  square  inch  is 


5500  X  5 
80  X  0.89  X 


=  49" 


To  avoid  the  use  of  so  heavy  a  slab  throughout  the  base,  a  fillet  of  con- 
crete will  be  placed  at  the  junction  of  the  base  and  counterfort,  dimensioned 
as  shown  in  Fig.  58.  The  main  body  of  the  slab  will  then  be  taken  as  2'  9" 
thick. 


FIG.  58. — Counterfort  wall. 

The  design  of  the  counterfort  proper  (note  that  a  final  check  of  the  dimen 
sions  just  found  is  omitted — in  actual  practice  such  omission  is  poor  design) 
is  most  conveniently  made  by  graphical  methods.  The  skeleton  outline 
of  the  truss  is  shown  in  Fig.  58.  The  loads  at  the  panel  points  A,  B,  C  are, 
allowing  for  the  ten  foot  spacing  of  counterforts : 

1.83  X  16      5  X  16 
Pa  -  2  — g- 

_  7_X _16   ,  5.5_XJ6      1^83_><  16  2X5  X  16  _ 

2                  6                     2  6 

12.5  X  15   ,  5.5  X  15   ,  7  X  16  ,  2  X  5.5  X  16 

ic  -       — 2 —             — A —              ~o —  — 6 —          = 

The  stress  polygon  is  drawn  as  shown  and  the  stresses  are  denoted  plus 
or  minus  as  they  are,  respectively  tension  or  compression.  The  vertical 


REINFORCED  CONCRETE  WALLS  119 

members    of  the  face  and  the  horizontal  member  'of  the  base,  must  carry 
the  moment  induced  by  the  slab  reactions.     These  moments  are 

4.3  X  162 

Mai  =  —  -  =  138  ft.  kips 

o 

Mbc  =  10  *  162  =  320  ft.  kips 


Mcd  = 

o 

The  unit  stress  in  tension  will  be  assumed  to  be  16,000  pounds  per  square 
inch.  That  in  compression,  long  column,  12,000  pounds  per  square  inch. 
The  vertical  arm  and  the  base  arm  are  buried  in  concrete.  It  is  the  practice, 
for  members  thus  stressed,  to  let  the  concrete  take  the  load  from  the  steel 
member  by  adhesion  so  that  the  member  carries  only  the  bending  load. 
Such  practice  will  be  adopted  here. 

Where  deductions  from  gross  section  are  necessary  because  of  rivet  holes, 
i^fe  inch  open  holes  will  be  assumed.  The  actual  work  of  the  design  is 
not  shown  here. 

ag.  S  =  61.  A  =    ^6  =    3.8  2  Ls  3.5  X  3.5  X  % 

db.  S  =  112.  A  =  ii%6  =7.  2  Ls  6  X  3.5  X  H 

be  S  =  -  125.  A  =  12^2  =  10.4  2  Ls  6  X  6  X  Ke 

eg  S  =  183  A  =  1%  =  H.4  2  Ls  6  X  6  X  %6 

cd  S  =  315  A  =  3i^6  =  19.7  2  Channels  15"  40# 

Since  the  member  AB  is  subject  to  bending  only, 

AB,  M  =  138  Sect.  Modulus     138  X  ^6  =  103 

Web  plate  15  X  %',  4  Ls  6  X  3,5  X  % 
EB  M  =  320.  S.  M.  =  320  X  %  =  240 

Web  plate  18  X  %;  4  Ls  6  X  6  X  ^6 
FD  M  =  424.  S.  M.  =  424  X  ^e  =  318 

Web  plate  24  X  %;  4  Ls  6  X  6  X  %. 

The  details  are  not  given  of  the  connections,  etc. 

It  will  be  assumed  that  the  truss  work  is  either  encased,  member  by  mem- 
ber in  concrete,  or  is  coated  with  gunite,  or  other  preparation  of  similar 
nature. 

6.  A  counterforted  wall,  24  feet  high,  subject  to  a  surcharge  of  6  feet,  is 
to  rest  upon  a  soil  capable  of  holding  not  more  than  5000  pounds  per  square 
foot.  Determine  the  general  wall  outlines  and  design  the  toe  extension. 

From  (95),  with  Si  =2.5  tons  and  H  =  30  feet,  and  i  =  e 


g  =  — 

From  Table  17,  for  e  =  0.25,  k  =  0.56,  and  the  width  of  the  base  is  0.56  X 
24  =  13'  6".  The  toe  projection  is  0.28  X  13.5  =  3.8  or  4'  0".  Without 
attempting  to  design  the  separate  sections  of  the  wall  and  then  redetermining 
these  general  outlines  from  the  more  exact  data,  let  it  be  assumed  that 
these  preliminary  outlines  will  remain  in  the  final  analysis. 


120  RETAINING  WALLS 

The  loading  upon  the  toe  extension  is  shown  in  Fig.  59.  R  =  30  X  9.5  X 
0.1  =  28.5  kips.  From  (39)  Si  =  4.9  kips  checking  the  first  assumption. 
From  (41),  the  location  of  the  point  of  zero  intensity  of  soil  pressure  is 

found  at  x  =  ™  !  ~~  H  =  4.5(0.16/0.44)  =  1.63  feet  from  the  heel.     The 

o    1   —  ZtC 

center  of  gravity  of  this  loading  may  be  found  by  aid  of  Table  3,  noting  that 
the  value  of  c  is  7.8/4.0  =  2  approx.,  whence  B  =  0.47  and  the  location  of 
the  force  is  1.88  from  the  toe,  little  error  would 
have  resulted  in  taking  the  center  of  gravity 
at  the  center  of  the  load.     The  total  load  is 

16.6.    For  shear  d  =  16,600/5040 

=  3.3.  The  moment  requirement  is  less  and 
the  depth  chosen  will  be  that  required  by  the 
shear.  The  total  thickness  of  the  toe,  includ- 
ing the  protective  concrete  over  the  steel  rods 
will  be  3'  6". 

_  16.600  X  2.12  X  12  _  OQ 
12  X  392 

and  pj  =  23/16,000  =  0.0014.  This  is  sub- 
FIG.  59.  stantially  the  steel  ratio  p.  The  area  of  steel 

required  becomes  fr.0014  X  39  X  12  =  0.66 

square  inches.  Taking  j  again  as  0.89,  the  periphery  of  steel  necessary 
for  the  proper  adhesion  stress,  namely  80  pounds  per  square  inch,  is 

16,600 


This  latter  requirement  controls  the  selection  of  the  reinforcement  and 
%  inch  square  bars  spaced  on  6"  centers  will  be  used.  Since,  to  develop 
the  stress  (and  in  accordance  with  the  principle  of  the  proper  detailing  of 
structures,  the  section  as  used  is  developed  and  not  merely  the  stress  exist- 
ing in  it)  the  bars  will  be  carried  by  the  face  of  the  vertical  arm  for  50  X 
%  =  4  feet. 

The  toe  as  finally  laid  out  is  shown  in  Fig.  59. 

It  must  be  again  emphasized  that  in  none  of  the  preceding  problems  have 
the  secondary  rod  systems,  for  temperature,  etc.,  been  shown.  In  a  later 
chapter  these  rod  systems  will  be  completely  detailed,  with  reference  to 
these  problems. 

Bibliography 

The  following  is  a  list  of  articles  on  reinforced  concrete  walls  : 
Standard  Design  of  5516  Linear  Feet  of  Wall,  9  to  24  Feet  in    Height, 

Steptoe  Smelter,  Engineering  Record,  Vol.  61,  p.  209. 
Recent  Retaining  Wall  Practice,  Journal  Western  Society  of  Engineers, 

Vol.  26. 
Tables  for  Reinforced  Concrete  Walls,  Based  on  Fluid  Pressures  of  20  and 

26.6  Pounds  per  Cubic  Foot,  Engineering  &  Contracting,  Vol.  xlii,  p.  146. 
Reinforced  Brickwork,  The  Engineer  (London,  England),  July  2,  1915. 


REINFORCED  CONCRETE  WALLS  121 

Design  of  Retaining  Walls,  Engineering  and  Maintenance  of  Way,  March, 

1912. 

Reinforced  Concrete  Retaining  Walls,  Cornell  Civil  Engineer,  March,  1913. 
Some  Economical  Types  of  Retaining  Walls,  Railway  Age  Gazette,  April  6, 

1917. 
Counterforted  Walls,  Lining  a  Stream  Channel,  Engineering  News,  Vol.  72, 

p.  1258. 
Walls  for  Yale  Bowl,  Maximum  Height  42  Feet,  Engineering  News,  Vol.  72, 

p.  997. 
Counterforted  Walls  with  Structural  Steel  Frame,  Engineering  News,  Vol. 

73,  p.  776. 
The  Design  of  Counterforted  Walls,  E.  GODFREY,  Engineering  &  Contracting, 

Vol.  xxxiv,  Dec.  21,  1910. 

(See  Also  Bibliography  in  Appendix.) 


CHAPTER  IV 
VARIOUS  TYPES  OF  WALLS 

The  types  of  walls  discussed  in  the  previous  chapters  are 
those  generally  used  in  engineering  practice.  Occasionally,  condi- 
tions are  such  that  these  general  types  are  inapplicable  and  it 
becomes  necessary  to  devise  special  types  to  meet  the  peculiari- 
ties of  the  given  environment.  Such  walls  are  described  briefly 
below. 

Cellular  Walls. — A  type  of  wall  insuring  a  light  foundation 
pressure  approaching  a  uniform  distribution  is  shown  in  Fig.  60. 
It  is  essentially  a  gravity  type,  the  interior  concrete  replaced  by 


Section  a- a 


Plan 


FIG.  60.— Cellular  wall. 


an  earth  fill.  The  principles  governing  its  outlines  are  thus  iden- 
tical with  those  governing  the  outlines  of  the  rectangular  gravity 
walls,  with  the  correct  allowance  made  for  the  reduced  stability 
moment.  In  a  finished  wall,  complete  with  the  fill  outside  and 
inside,  the  rear  wall  is  under  no  pressure.  To  insure  no  possi- 
bility of  failure  during  construction  or  at  some  later  date  in  con- 
sequence of  an  adjacent  excavation,  it  is  well  to  make  the  rear 
wall  like  the  face  wall.  Theoretically  the  wall  may  be  built 
without  a  base.  Practically,  to  insure  an  even  distribution  of 
pressure  upon  the  bottom,  and  to  avoid  unsightly  settlement,  a 
base  is  generally  used. 

The  design  of  the  separate  members  is  identical  with  the 
method  used  in  the  design  of  the  several  members  composing 
the  counterf  orted  wall.  For  the  base,  when  such  is  used,  the  slab 
should  be  designed  for  the  net  difference  between  the  upward 

122 


VARIOUS  TYPES  OF  WALLS 


123 


and  downward  loads.  A  description  of  a  wall  of  this  type  is 
given  in  Engineering  &  Contracting,  Vol.  35,  p.  530,  by  J.  H.  Prior. 
Hollow  Cellular  Walls. — To  insure  even  lighter  soil  pressures 
than  given  by  the  type  previously  discussed,  a  hollow  cellular 
wall  may  be  used,  as  described  in  Fig.  61.  Its  stability  is 


Section  a-a  Plan 

FIG.  61. — Hollow  cellular  wall. 

furnished  by  the  small  amount  of  earth  fill  resting  immediately 
upon  it  and  by  the  weight  of  the  track  ballast,  in  addition  to  the 
weight  of  the  separate  members  composing  the  cells.  It  is 
essential,  because  of  the  light  weight  of  the  wall  that  adequate 
attention  should  be  paid  to  its  tendency  to  slide  forward.  The 
face  of  the  lower  part  of  the  wall  should  abut  against  the  firm 
ground,  and,  if  possible,  extensions  should  be  built  into  the  bot- 
tom to  add  to  the  sliding  resistance.  Two  interesting  types 
of  the  wall  are  described  here.  The  former,  as  shown  in  Fig.  61, 


V            , 

\> 
i  Q 

/                             ™— 

'"  Openings  in 
Partition  Walls 

•* 

Lfr           '    S>«4  -  3  *            J 

p;            ct-  o     —  T\ 

aj 

Beam  Struts 


...-  Slope  U'ne  of 
Pressure 


FIG.  62. — Cellular  wall  on  timber  cribbing. 

termed  the  "Lacher"  wall  is  described  in  detail  in  an  article  by 
J.  H.  Prior.1  While  this  was  the  most  expensive  type  of  five 
types  analyzed  for  the  track  elevation  work  of  the  Chicago, 
Milwaukee  and  St.  Paul  (gravity,  "L"  shape,  counterfort  "L," 
cellular  as  described  previously  and  the  hollow  cellular)  it  was 
the  only  type  insuring  a  safe  permissible  pressure  on  the  soil 
encountered  in  the  work.  The  maximum  soil  intensity  was  two 
1  Engineering  &  Contracting,  Vol.  35,  p.  530. 


124  RETAINING  WALLS 

tons  per  square  foot.  This  type  also  permitted  a  full  use  of  the 
easement  for  tracks.  It  was  not  feasible  to  use  piles. 

The  second  type,  shown  in  Fig.  62,  was  used  in  supporting  the 
Speedway,  a  highway  along  the  west  bank  of  the  Harlem  River, 
New  York  City.  It  is  described  in  the  Engineering  Record,  Vol. 
66,  p.  22.  A  good  foundation  could  be  had  upon  a  timber 
cribbing  already  in  place,  below  mean  high  water,  giving  promise 
of  little  future  settlement.  The  wall  is  about  square  in  section 
and  the  sidewalk  forms  the  upper  slab  of  the  cell.  The  walls 
are  thinned  down  towards  the  top  and  a  circular  segment  is  cut 
out  of  the  transverse  walls,  to  diminish  the  load  upon  the  base. 
The  distribution  of  the  pressure  is  practically  a  uniform  one. 
To  quote  from  the  article : 

"The  transverse  walls  are  so  spaced  that  their  weight  is  evenly  dis- 
tributed upon  the  foundation  cribs  by  the  3  foot  concrete  flooring. 
It  was  assumed  that  the  line  of  thrust  at  the  base  of  these  walls  due  to 
their  weight  and  the  weight  of  the  sidewalks  which  they  carry,  would 
be  at  an  angle  of  45°.  Upon  this  basis,  the  lines  of  thrust  from  the 
bottoms  of  successive  transverse  walls  intersect  just  at  the  base  of  the 
3  foot  concrete  floor,  causing  a  uniform  application  of  the  loads  upon 
the  foundation  cribs."  See  Fig.  62. 

Timber  Cribbing. — Walls  have  been  constructed  of  old  ties, 
forming  practically  cellular  walls.  The  transverse  ties  are 
spiked  to  the  stretcher  ties  forming  the  rear  and  front  faces.  See 
Fig.  63.  Such  a  wall  was  used  in  Chicago  by  the  Chicago,  Rock 


FIG.  63. — Timber  crib. 

Island  and  Pacific  Railroad  for  heights  varying  from  four  to 
twenty  feet.  There  is  an  interesting  discussion  on  the  use  of  this 
type  of  wall  in  the  Joural  of  the  Western  Society  of  Engineers, 
Vol.  20,  232  et  seq. 

Concrete  Cribbing. — In  exactly  identical  fashion  with  the  use 
of  timber  cribs,  concrete  cribbing  may  be  used,  the  members 
constructed  in  units  of  a  shape  similar  to  a  tie  and  reinforced  at 
the  four  corners.  A  description  of  the  use  of  such  cribbing  in 
Oregon  along  a  highway  is  given  in  the  Engineering  News- 
Record,  Vol.  81,  p.  763.  It  is  pointed  out  in  this  article  that  the 


VARIOUS  TYPES  OF  WALLS 


125 


life  of  timber  cribs  is  so  short  that  their  use  is  not  economical. 
Concrete  cribs,  would  not  be  open  to  this  objection. 

Walls  with  Land  Ties  (or  Backstays). — This  is  a  practically 
obsolete  type  of  wall,  but  is  occasionally  used  for  small  light 
walls  usually  along  the  water  front.  A  typical  wall  of  such  charac- 
ter is  described  in  Engineering  and  Contracting,  Voi.  37,  p.  328. 
It  is  shown  in  Fig.  64.  Its  design  follows  from  the  ordinary 


•9-2* 


FIG.  64.— Wall  with  land  ties. 

principles  of  statics  and  the  force  system  is  shown  in  Fig.  64. 
If  the  tie  is  a  metal  one,  there  is  danger  of  its  gradual  destruction 
by  rust.  It  should  be  encased  in  concrete,  which  adds  consider- 
ably to  the  expense  of  the  wall.  On  a  fair  foundation  and  for  a 
small  wall,  this  type  may  prove  economical.  The  theory  of  such 
walls  is  given  by  Rankine  23rd  Ed.,  1907,  pp.  410,  411. 

Walls  with  Relieving  Arches.— This  is  another  type  of  his- 
torical interest  rarely  used  now.     As  constructed  of  brick  with 




jfnnniiini\ 


FIG.  65. — Wall  with  relieving  arches. 

cheap  labor  it  afforded  an  economical  type  of  substantial  con- 
struction. The  theory  of  such  a  wall  is  given  by  Rankine,  in 
his  23rd  Ed.,  p.  412.  Fig.  65  shows  a  typical  view  of  such  a 
wall. 

An  interesting  example  of  a  wall  of  this  kind  is  given  on  p. 
353  Handbuch  Fiir  Eisenbetonbau  III  Band.    The  relieving  arches 


126 


RETAINING  WALLS 


are  of  cast  iron  and  the  wall  masonry  of  brick.     The  section  of 
the  wall  is  shown  in  Fig.  66. 

A  novel  type  of  wall  is  shown  in  Fig.  67,  and  is  a  compromise 
between  a  cellular  and  cantilever  type.  It  is  taken  from  the 
handbook  on  concrete  quoted  above. 


FIG.  66. — Brick  wall  with  cast  iron  relieving  arches. 


FIG.  67. — Special  shape  wall. 

Euorpean  Practice. — Some  very  interesting  types  of  walls, 
m  ostly  of  European  origin  are  given  in  the  Handbuch  Fur  Eisen- 
betonbau  III  Band,  pp.  369  to  402.  The  intricate  rod  systems 
and  complicated  form  details  necessary  in  the  construction  of 
these  walls  would  preclude  their  use  in  America.  It  is  notable 
to  see  the  latitude  allowed  individual  engineering  talent  in  the 
adoption  of  the  various  designs  and  such  freedom  of  thought 
should  prove,  in  the  long  run,  very  fruitful  in  useful  wall  sections. 

Embankments  Bounded  by  Two  Walls. — The  construction  of 
embankments  through  narrow  easements,  requiring  retaining 
walls  on  either  side  of  the  fill  makes  it  possible  to  utilize  the 
mutual  action  of  the  two  walls  to  effect  quite  a  reduction  in  the 
section  of  each  wall  required.  The  wall  thus  built  is  in  effect 
a  modification  of  the  counterforted  wall  and  so  far  as  the  actual 
design  of  the  wall  itself,  the  theory  as  previously  given  is  sufficient 


VARIOUS  TYPES  OF  WALLS 


127 


FIG.  68.— Walls  of  Hell  Gate  arch 
approach. 


to  design  this  wall.     Two  interesting  examples  of  this  type  of 
construction  are  given  here. 

RETAINING  WALLS,  NEW  YORK  CONNECTING  RAILROAD, 
HELL  GATE  ARCH 

Approach. — 'The  embankment  to  be  retained  was  practically 
of  square  section,  60  feet  wide  and  high.  The  ordinary  theory 
of  earth  pressure  would  have  necessitated  enormous  sections.  A 
carefully  specified  embankment 
well  drained  and  compacted 
made  it  possible  to  reduce  the 
thrusts  (see  page  21).  The 
walls  were  divided  into  ten  foot 
square  panels,  at  each  corner  of 
which  a  tie  rod  2^  inch  diameter 
extended  between-  the  walls  and 
was  anchored  to  a  steel  channel 
embedded  in  the  face  walls  (see 
Fig.  68).  Every  fifty-feet,  a 
partition  wall  ran  between  the  face  walls  giving  additional  stabil- 
ity to  the  section,  and  especially  stiffness  against  wind  stresses 
prior  to  the  placing  of  the  fill  within  the  wall.  A  most 'careful 
system  of  drainage  was  placed  at  every  row  of  tie  rods  to  prevent 
the  accumulation  of  water  with  a  consequent  increased  pressure. 

INTERBORO    RAPID    TRANSIT    RAILROAD,    EASTERN   PARKWAY 

IMPROVEMENT 

The  walls  here  were  about  25  feet  high  and  tied  to  each  other 
at  intervals  of  20  feet  by  reinforced  concrete  partition  walls 
(see  Fig.  69). 

In  both  examples  it  is  to  be  noticed  that  no  bottom  slab  is 
used,  forming  the  true  cellular  wall  as  described  by  Lacher  in 
the  previously  mentioned  issue  of  the  Journal  of  the  Western 
Society  of  Engineers.  The  interesting  details  in  connection 
with  the  use  and  non-use  of  expansion  joints  are  discussed  in  the 
following  chapter. 

The  widening  of  an  existing  right  of  way  prior  to  its  final  com- 
pletion (White  Plains  Rd.  Extension,  Interboro  Rapid  Transit 
Co.)  made  it  possible  to  adopt  an  unusual  expedient  of  anchoring 
the  new  wall  directly  to  the  existing  wall.  Structural  steel 


128 


RETAINING  WALLS 


frames  were  anchored  through  the  existing  wall  as  shown  in  Fig. 
70  (See  Plate  II,  Fig.  26).  The  new  face  wall  consisted  of 
slabs  supported  by  upright  channels.  To  insure  the  permanence 
of  the  anchors  they  were  embedded  in  concrete  partition  walls. 
In  placing  the  fill  care  was  observed  to  carry  up  the  fill  levels  at 
the  same  rate  on^either  side  of  these  partition  walls  to  prevent 


4' 


j 


Plan 


Section  a-a 
FIG.  69. — Walls  Eastern  Parkway  Extension  Interboro  Rapid  Transit  R.  R. 

placing  an  earth  pressure  upon  them.  The  thickness  of  the 
face  slabs  was  the  minimum  width  it  was  found  practicable  to 
construct  in  the  field  with  the  equipment  at  hand. 

Abutments. — The  design  of  the  abutment  differs  from  that  of 
the  ordinary  retaining  wall,  merely  in  that  an  extra  dead  or  dead 
and  alive  load,  is  superimposed  upon  the  wall  and  serves  to 
counteract  the  overturning  moment  of  the  earth  pressure.  This 


Plan 


Section  a-a 

FIG.  70. — Anchoring  new  wall  to  old  wall. 

additional  load,  resting  upon  the  abutment  is  assumed  to  be 
uniformly  distributed  along  the  abutment  and  is,  thus,  treated, 
mathematically,  as  an  additional  masonary  surcharge.  The 
variable  conditions  of  loading  make  it  necessary  to  investigate 
all  possible  states  of  loading,  in  order  to  ascertain  the  maximum 
forces  upon  the  wall. 

The  following  combinations  of  dead  and  live  loads  are   all 
possible  ones  and  each  is  worthy  of  investigation.     The  ac- 


VARIOUS  TYPES  OF  WALLS  129 

company  ing  Fig.  71  may  serve  to  give  a  better  idea  of  these 
combinations  as  listed  below. 

(a)  The  earth  backing  in  place,  but  no  span  construction  set. 
The  abutment  is  a  plain  retaining  wall. 

(b)  The  crane  to  be  used  in  erecting  the  span  is  in  place  behind 
the  abutment.     Here  the  abutment  is  a  retaining  wall  with  a 
surcharge  load  due  to  the  erecting  crane. 

(c)  The   construction  complete.    Live  load  approaching  the 
span.     The  abutment  has  the  full  earth  and  surcharge  load,  but 
only  the  dead  load  of  the  span  as  a  relieving  load. 


(c)  (d) 

FIG.  71. — Conditions  of  Abutment  loading. 

(d)  The  live  load  is  on  both  the  span  and  back  of  the  abutment. 
There  is  here  the  maximum  earth  pressure  and  maximum  relief. 
This  latter  case  gives  the  greatest  total  loading  upon  the  base. 
The  others,  however,  may  give  a  greater  toe  intensity. 

In  connection  with  the  conditions  of  loading  subsequent  to 
the  completion  of  the  structure,  the  span  construction,  in  ad- 
dition to  the  relief  afforded  by  its  weight  upon  the  wall  also 
exerts  a  horizontal  relieving  action,  forming  a  beam  out  of  the 
abutment  with  both  a  top  and  bottom  support.  Such  relief, 
however,  is  most  difficult  to  compute,  due  to  the  uncertainty 
of  the  action  of  the  roller  bearings  and  had  better  be  neglected 
in  the  design  of  the  wall. 

The  designer  should,  of  course,  govern  the  design  of  the  wall 
by  the  above  four  conditions  and  not  attempt  to  control 
the  field  conditions,  such  as  the  sequence  of  operations  in  the 
placing  of  embankment  and  erection  of  the  bridge,  by  his  design. 
It  is,  of  course,  within  the  province  of  the  experienced  engineer 
to  determine  how  best  to  adapt  the  design  to  take  care  of  the 


130 


RETAINING  WALLS 


construction  loadings.  The  factor  of  safety  against  sliding  and 
overturning  may  be  temporarily  lowered  to  take  into  account  the 
conditions  prior  to  final  completion,  but  it  does  not  seem  advis- 
able to  permit  the  soil  intensity  under  any  combination  of 
loading,  temporary  or  otherwise,  to  exceed  the  safe  allowable 
pressure. 


FIG.  72. 


FIG.  73. 
Abutment  types. 


FIG.  74. 


The  location  of  an  abutment  is  usually  transverse  to  the  right 
of  way,  permitting  the  footing  to  encroach  upon  the  crossing, 
whether  public  or  private.  It  is  thus  possible  to  secure  the  best 
type  of  soil  pressure  distribution,  keeping,  at  the  same  time,  an 
economical  section  of  wall.  Since  the  abutment  is  a  combination 
of  a  retaining  wall  and  an  ordinary  pier  subject  to  vertical  loads 
only,  it  is  customary  to  extend  both  the  heel 
and  toe  (see  Figs.  73,  74,  75). 

Abutments  may  be  either  composed  of  plain 
masonry  or  ol  reinforced-concrete,  as  economy 
or  other  factors  dictate.  The  flexibility  of 
reinforced-concrete  in  permitting  slender  walls 
with  projecting  heel  and  toe  indicates  that  for 
practically  every  condition  a  reinforced-concrete 
type  of  wall  may  be  found  that  will  prove  more 
economical  than  the  gravity  masonry  walls. 
The  counterforted  retaining  walls  may  readily  be  adapted  to 
form  an  abutment,  by  placing  a  cap  over  the  top  to  form  the 
girder  seat  (see  Fig.  72).  Several  of  the  usual  types  of  abut- 
ments are  shown  in  Figs.  73,  74  and  75. 

Wing-walls. — The  wing  walls  attached  to  the  abutments  are 
ordinary  retaining  walls  and  are  so  designed.  Their  location 
is  governed  by  the  conditions  of  the  intersection  and  may  either 
be  in  line  with  the  abutment,  following  the  slope  of  the  fill,  or 


FIG.  75. — Re- 
inforced-concrete 
abutment. 


VARIOUS  TYPES  OF  WALLS  131- 

if  the  condition  of  the  easement  does  not  permit  may  make 
an  angle  with  the  abutment  determined  by  the  economical 
limitations.  The  combination  of  wing  wall  and  abutment,  makes 
it  possible  to  devise  ingenious  schemes  to  effect  an  economy  of 
material  used.  The  walls  and  abutment  may  form  a  U-band  of 
constant  cross-section  as  described  in  Engineering  News,  Mar.  8, 
1917,  p.  393,  the  walls  partially  buried  in  the  fill  and  holding,  by 
friction,  the  abutment  portion  of  the  U.  Cellular  abutments 
have  also  been  used. 

Occasionally  an  abutment  is  supported  by  a 
stem  buried  in  the  retained  embankment,  forming 
a  T  (see  Fig.  76). 

An  exhaustive  analysis  of  abutments  and  wing 
walls,  with  a  wealth  of  practical  hints,  is  given  by 
J.  H.  Prior  in  the  American  Railway  Engineering 
Association,  Vol.  13,  p.  1085. 

C.  K.  Mohler,1  Consulting  Engineer,  has 
pointed  out  the  economy  effected  by  turning 
back  the  wing  wall  in  place  of  merely  extending  it  in  the  line  of 
the  abutment  to  follow  the  slope  of  the  retained  embankment. 
E.  F.  Kelly  has  pointed  out2  that  for  minimum  wing  length, 
the  face  of  the  wing  should  bisect  the  angle  between  the 
shoulder  of  the  fill  (sometimes  termed  the  berm)  and  the 
face  of  the  abutment  produced.  This  assumes  that  the 
end  of  the  wing  wall  becomes  a  line,  in  place  of,  as  in 
actual  practice,  the  wall  being  cut  off  at  a  convenient  height. 
Since  the  end  of  the  wall  has  no  serious  effect  upon  the  entire 
amount  in  question,  such  approximation  has  but  negligible 
effect.  To  take  into  account  such  practical  factors,  the  author 
of  the  paper  has  prepared  curves  giving  the  actual  angle  required 
when  the  character  of  the  end  detail  is  taken  into  account  to- 
gether with  the  character  of  the  junction  of  the  wing  with  abut- 
ment at  the  shoulder.  It  is  emphasized3  that  where  minimum 
volume,  rather  than  minimum  length  is  sought,  the  above  rule 
and  curves  do  not  hold.  For  minimum  volume  the  wing  wall 
carried  out  directly  in  the  plane  of  the  abutment  face  gives  the 
least  volume  until  the  angle  between  the  wing  and  the  axis  of 
the  retained  embankment  exceeds  a  right  angle. 

1  Engineering  News-Record,  Vol.  80,  p.  168. 

2  Ibid,  p.  785. 

3  Ibid,  p.  1243. 


132  RETAINING  WALLS 

For  track  elevation,  where  full  trackage  on  a  limited  easement 
is  essential,  the  abutment  frames  into  the  two  parallel  retaining 
walls  on  either  side  of  the  embankment  forming  a  box-like 
structure.  Other  details  are  made  to  fit  into  the  special  cir- 
cumstances of  the  given  location. 

A  number  of  examples  of  the  varied  types  of  gravity  and  re- 
inforced concrete  abutments  is  given  in  the  Handbuch  fur 
Eisenbetonbau  iii  Band,  pp.  415  to  422. 

For  ordinary  highway  abutments  it  is  possible  to  compile 
standard  sections  to  cover  practically  all  the  cases  expected. 
Thus  H.  E.  Bilger  in  a  paper  read  before  the  Illinois  Society 
of  Engineers  and  Surveyors1  states:  For  walls  up  to  25  feet  in 
height : 

( a)  For  ordinary  earth  bottoms,  the  base  is  Y%  the  height ; 

(b)  For  rock  or  shale  bottoms  the  base  is  J4  the  height. 

The  footing  is  18  inches  thick  and  is  offset  9  inches  at  the  heel 
and  toe.  The  back  of  the  wall  is  vertical.  Gravity  walls 
are  generally  used  because  the  character  of  local  labor  does 
not  permit  the  use  of  the  reinforced  concrete  sections. 

Box  Sections  Subject  to  Earth  Pressures.— The  section,  shown 
in  Fig.  77,  subjected  to  earth  pressure,  both  horizontal  and  ver- 
tical requires  an  intricate  analysis,  if  de- 
_Surface signed  as  a  monolith.     Since  such  struc- 
tures,    though     otherwise    designed,    are 
actually  rigid  frames,  it  is  quite  desirable  to 
-  nfa   learn  the  true  stresses  existing  in  them. 

The  principles  of  the  theory  of  least 
work  applicable  to  the  problem  in  question 
may  be  stated  as  follows: 

(a)  The  work  performed  by  the  shear 

FIG  77. — Sub-surface  T.I          .    •  T    -i  i  •  -^i 

structures  an^  thrust  is  negligible  in  comparison  with 

the  work  done  by  the  moment. 

(b)  The  work  performed  by  the  moment  between  any  two 
points  Si  and  s2  is  given  by  the  expression: 

*s*M*dx 


(147) 

(c)  The  derivative  of  this  expression  with  respect  to  a  force 
that  does  no  work  i.e.,  a  force  whose  point  of  application  is  at  a 
fixed  point,  is  zero. 

1  Given  in  Engineering  Record,  Vol.  63,  p.  205. 


VARIOUS  TYPES  OF  WALLS 


133 


Corollary :  It  is  permissible  to  differentiate  the  expression  under 
the  integral  sign  with  respect  to  a  variable  other  than  the  variable 
of  the  integrand,  thus 

*St 

(148) 


-!' 

JSi 


Finally,  it  shall  be  arbitrarily  taken  that  a  moment  which  causes 
compression  in  the  outside  of  the  member  is  positive. 

In  Fig.  78  the  moments  between  the  following  points  are: 
C  to  a:  M  =  -Ml  +  Ex 
a  to  A :       =  -Ml-  W(x  -  a)  +  Hx      • 
A  to  B:      =  -Mi  -  W(h  -  a)  +  Hh 
B  to  D:      same  as  c  to  A 

The  total  work  is,  with  /i  and  72  the  moments  of  inertia  of  the 
roof  and  sidewalls  respectively,  and  E  the  modulus  of  elasticity 


=  JL|  \     (- 


w  = 


Hx^dx  + 


f  [  -Mi  - 


W(x  -  a)  +  Hx]*dx 


W 


M,      H 


W 


FIG.  78.  FIG.  79. 

Loads  on  sub-surface  frame. 


The  forces  M  i  and  H  shall  be  taken  as  the  forces  with  respect 
to  which  the  partial  derivatives  of  the  work  are  zero.     The  points 
C  and  D  are  taken  as  fixed.     From  the  corollary  and  since 
dw/dH  =  dw/dMl  =  0 


1 
El, 


-Ml  +  Hx)xdx 


f. 

Ja 


-Ml-  W(x  -  a) 


Hx]xdx  \ 


1 
W, 


2[~Ml    W(h  ~a)  +  Hh]hdx\  =  ° 


134  RETAINING  WALLS 

-  2[-  Jfi  -  W(x-  a)  +  Hx]dx 

-2[-Mi-W(h-d)+Hh]dx\  =  0 

Solving  these  two  simultaneous  equations  for  H  and  MI 
M        Wa(h  -  q)[AJ,(fc  -  q)  +  6/2(2ft  -  a)] 
Ml=  "'"     '   2bJ2) 


„       W(h  -  a)*[h(h  +  2a)Ij  +  6(2ft  +  a)/2]         ,      , 

AWi  +  26/0 

In  similar  fashion,1  referring  to  Fig.  79,  the  base  moment  and 
horizontal  thrust  due  to  concentrated  load  upon  the  roof  is  found 


Using  these  four  equations  as  a  foundation,  it  is  possible  to 
establish  some  general  conditions  of  loading  on  either  roof,  side- 
walls  or  upon  both  simultaneously.  For  a  uniformly  distributed 
load  on  the  roof  of  w  per  foot,  replace  in  (152)  a  by  x,  W  by  w, 
multiply  the  expression  by  dx  and  integrate  between  the  limits  0 
and  b.  The  expressions  for  the  thrust  H  and  the  mo  ent  M  i 
are 


"  4h  Mi  +  26/2 

,,        wb2        6/2  f  „. 

JflS!l2M1  +  26/a 

For  a  uniformly  distributed  loading  p  on  the  side  walls,  in  similar 
manner  integrate  the  expressions  given  in  (150)  and  (151)  be- 
tween the  limits  0  and  h.  The  thrust  and  moment  are  then 


(156) 

4    ft/1 


,,  ph*  hlj  +  36/2  n  7v 

Ml=  "12  WTT26T,  (157) 

Again  for  a  triangular  distribution  of  loading  on  the  side  wall, 
with  maximum  base  intensity  q,  the  expressions  become 


qh  7hll  +  166J2  (       . 

20     M, 


-         60   W,  +  26/2 

1See  HIROI,  "Statically  Indeterminate  Structures." 


VARIOUS  TYPES  OF  WALLS 


135 


Denote  the  ratio  ^  by  e  and  let  1/(1  +  2e)  =  Z\\  (2  +  5e)/ 
till 

(1  +  26)  =  Z2.  Then  (1  +  3e)/(l  +  2e)  =  Z,  -  1;  (7  +  16e)/ 
(1  +  2e)  =  3  +  2Z2;  (3  +  8e)/(l  +  2e)  =  1  +  2Z2.  Table  25 
gives  the  values  of  Z\  and  Z2  for  several  values  of  the  ratio  e. 

With  the  above  substitutions  the  expressions  given  in  (154  to 
159)  become 


For  uniform  loading  on  roof. 


wb2 
"12" 


For  uniform  loading  on  side  wall. 

ph  „  _  _ 

T  Z2,  M: 


H 


For  triangular  loading  on  side  wall. 
H  =  20  (3  +  Z2)' 


ph? 
12 


=    -  ^-  (1   +   2Z2) 


(160) 


(161) 


(162) 


TABLE  25 


To  apply  these  expressions  to  a  sub- 
surface structure  subject  to  earth  pres- 
sure upon  roof  and  sidewalls,  let  the 
loading  above  the  roof  line  be  treated  as 
a  surcharge,  with  the  usual  terminology 
that  c  is  the  ratio  of  this  surcharge  height 
to  the  full  wall  height  h.  The  roof  load- 
ing w  is  then  gch  and  the  side  wall 
pressure  is  compounded  of  a  uniform 
intensity  p  =  Jgch  at  the  top  of  the  side 
wall,  and  a  triangular  loading  with  base 
intensity  q  =  Jgh.  For  a  loading  upon 
the  roof  alone  the  respective  thrust  and  moment  are 

H  =  ^Zt 
4 

TUT     ._   QdMy 


e 

Zi 

Zt. 

0 

1.00 

2.00 

.2 

.72 

2.14 

.4 

.56 

2.22 

.6 

.45 

2.27 

.8 

.38 

2.31 

1.0 

.33 

2.33 

1.5 

.25 

2.37 

2.0 

.20 

2.40 

Infin. 

.0 

2.50 

(163) 
(164) 


For  a  loading  upon  the  side  wall  alone  the  thrust  and  moment  are 


H 


Jgh2 
20 


[3  +  (2  +  5c)Z2] 


Jgh* 
60 


-  5c  +  (2  +  5c)Z2] 


(165) 
(166) 


136  RETAINING  WALLS 

For  a  simultaneous  load  upon  roof  and  sidewall  the  two  above 
expressions  are  added  to  give  the  total  thrust  and  moment.  It 
is  possible,  of  course  to  have  a  different  surcharge  for  the  roof 
than  for  the  sidewall,  since  there  may  be  no  surface  load  over  the 
roof  and  a  surface  load  whose  weight  will  affect  the  sidewall  pres- 
sure. This  is  taken  care  of  by  giving  the  proper  values  to  the 
surcharge  ratio  c  in  the  above  expressions. 

With  the  thrust  H  and  the  base  moment  MI  known  the  moment 
at  any  other  point  of  the  frame  can  easily  be  found  by  the  ordi- 
nary principles  of  statics. 

Fig.  89  is  a  typical  section  of  such  a  structure  analyzed  by 
the  above  method.  A  radically  different  distribution  of  stress 
exists  in  this  structure  when  analyzed  exactly  as  above  than 
when  it  is  treated  as  an  assembly  of  independent  units.  It  is 
the  very  essence  of  the  design  of  such  structures,  usually  subsur- 
face, that  they  be  waterproof.  Any  cracks  developed  in  the 
structure  due  to  ignored  stresses  are  fatal  to  the  integrity  of  the 
structure.  It  is  patent  that  regardless  of  what  method  is  em- 
ployed in  designing  such  structures,  provision  must  be  mad  •  for 
stresses  as  found  above. 

The  theory  as  above  outlined  and  the  formulas  as  given  are 
ample  to  analyze  any  subsurface  structure  subject  to  lateral  and 
vertical  pressures. 

The  mutual  effect  of  the  members  upon  each  other  makes  it 
essential  that  such  conditions  be  combined  as  will  produce  the 
maximum  stresses  at  the  separate  points  of  the  structure. 

It  may  be  interesting  to  note,  while  treating  sub-surface  struc- 
tures that  a  very  thorough  analysis,  both  theoretical  and  practical, 
of  stresses  in  large  sewer  pipe  is  given  in  Bulletin  No.  31,  issued 
by  the  Engineering  Experimental  Station  of  the  Iowa  State 
College  of  Agriculture  and  Mechanic  Arts.  See  also  for  a  com- 
parison between  theoretical  and  actual  stresses  "  Analysis  and 
Tests  of  Rigidly  Connected  Reinforced-Concrete  Frames"  by 
Mikishi  Abe,  Bulletin  No.  107.  Engineering  Experiment 
Station,  University  of  Illinois. 

Economy  of  the  Various  Types. — -Broadly  speaking,  the  selec- 
tion of  a  given  type  of  wall  is  governed  by  one,  or  more  of  the 
following  reasons:  economy  of  section;  character  of  foundation; 
demands  of  the  environment,  in  which  latter  may  be  included 
the  relation  between  walls  and  property  line;  architectural 
treatment,  the  wall  entering  into  a  part  of  some  general  landscape 


VARIOUS  TYPES  OF  WALLS  137 

scheme;  the  availability  of  materials  necessary  for  its  construction 
and  the  character  of  the  labor  to  be  had  in  the  vicinity  of  the 
work. 

So  far  as  the  economy  of  the  section  is  involved,  it  must  be 
noted  that  the  relative  economy  of  gravity  and  reinforced  con- 
crete walls  is  not  that  given  merely  by  a  parallel  comparison  of 
materials  required  for  the  finished  wall.  The  reinforced  concrete 
wall  has  thinner  members,  requiring  more  form  work  per  cubic 
yard  of  -concrete.  The  slenderness  of  this  wall,  together  with 
the  net-work  of  rods  within  it,  makes  it  more  difficult  to  properly 
place  and  distribute  the  concrete,  necessitating  more  skillful 
labor  and  more  competent  foremanship.  The  gravity  walls 
are  more  capacious  within  the  forms,  the  laborers  have,  conse- 
quently, more  room  to  move  about  and  can  thoroughly  spade 
and  turn  over  the  mix,  giving  better  assurance  of  a  flawless  wall. 
This  is  a  very  important  item  and  one  too  frequently  overlooked. 
A  concrete  gang  of  the  average  type,  i.e.,  a  class  of  men  just  a 
shade  above  the  common  excavators,  will  tackle  a  gravity  section 
of  wall  and  turn  out  a  good  looking  section.  Upon  attempting 
to 'pour  a  reinforced  concrete  wall,  a  very  inferior  piece  of  work 
is  constructed.  Before  preparing  plans  for  a  thin  reinforced 
concrete  wall,  it  is  essential  to  insist  upon  a  capable  contractor, 
equipped  with  the  proper  labor  gangs  to  do  such  work.  With 
a  policy  of  awarding  the  work  to  the  lowest  bidder  where  competi- 
tive bids  are  asked,  it  is  necessary  that  the  engineer  adapt  the 
type  of  wall  to  one  that  can  safely  be  built  by  the  general  run  of 
low  bidders. 

Unsuspected  variations  in  the  character  of  foundations,  may 
demand  an  abrupt  change  in  the  section  of  wall.  For  a  rein- 
forced concrete  wall  the  rods  are  usually  ordered  some  time  in 
advance  of  the  actual  construction  of  the  wall.  It  is  necessary 
that  the  section  of  the  wall  be  determined  at  the  time  of  ordering 
the  rods1.  Despite  careful  boring  made  at  the  site  of  the  work, 
the  soil  encountered  at  the  proposed  bottom  of  the  wall  may  prove 
to  be  different  from  that  assumed  and  it  may  thus  become  neces- 
sary to  excavate  deeper  to  obtain  the  desired  character  of  bottom, 
or  even  to  change  the  type  of  wall.  Since  the  rods  have  been 
ordered,  the  wall  design  is  inflexible  and  if  a  new  section  is 

1  While  it  is  possible  to  get  shipments  from  local  markets  at  short  notice, 
quite  a  premium  must  be  paid  for  this  material  and  such  orders  are  given 
only  when  economy  must  be  sacrificed  to  urgency. 


138  RETAINING  WALLS 

ordered,  it  may  mean  delay  awaiting  mill  shipments  of  the  new 
lengths  needed,  costly  orders  of  rods  from  stock  supplies,  the  un- 
desirable splicing  of  rods  or  the  placing  of  a  plain  concrete  base 
to  bring  the  actual  bottom  level  up  to  the  theoretical  one — all 
expensive  and  undesirable  expedients.  For  this  condition  the 
gravity  wall  is  the  more  flexible  type  and  the  section  may  be 
changed  without  any  additional  trouble  should  soils  at 
variance  with  the  originally  assumed  ones,  be  encountered. 

On  the  other  hand,  where  the  character  of  the  soil  is  assured, 
the  reinforced  concrete  type  of  wall  may  be  molded  to  adapt 
themselves  to  any  distribution  of  soil  pressure  desirable.  This 
has  been  shown  in  the  previous  work. 

It  has  been  pointed  out1  .  .  .  for  walls  of  the  height  re- 
quired for  track  elevation  and  track  depression  a  gravity  wall, 
will  under  ordinary  conditions  be  cheaper  than  the  reinforced 
concrete  types. 

Again,  in  the  same  issue  of  the  Journal  in  discussing  the  relative 
demerits  and  merits  of  the  cellular  types  it  was  pointed  out2  in 
connection  with  track  elevation  work,  that  such  a  wall,  with  the 
bottom  left  out  offers  great  resistance  to  sliding  and  overturning 
and  "  occupies  the  right  of  way  so  as  to  afford  little  opportunity 
for  encroachment.  It  permits  of  ready  driving  of  a  pile  trestle 
right  over  it."  On  the  other  hand  "it  occupies  considerable 
space  before  filling  and  may  thus  interfere  with  the  use  of  the 
tracks.  Settlement  may  also  give  an  unpleasing  appearance." 

So  far  as  the  actual  amounts  of  materials  involved,  both 
during  construction  (forms,  etc.)  and  in  the  permament  structure 
it  is  possible  to  determine  the  more  economical  wall  by  com- 
parison of  two  types  or  by  mathematical  and  tabular  methods  as 
given  at  the  end  of  this  chapter.  It  is  understood  that  the  proper 
weight  is  given  to  the  indeterminate  factors  of  cost  as  above 
mentioned  i.e.  the  construction  limitations  of  the  several  types. 

It  must  be  emphasized  that  wall  details  should  be  simple. 
Shapes  that  apparently  make  for  economy  may  prove  exceedingly 
difficult  to  pour  in  the  field.  Thus  for  example,  a  section  of  a 
cantilever  wall  as  shown  in  Fig.  80  (see  also  Photo  Plate  No.  4a) 
with  a  net  work  of  obstructing  rods  at  A  makes  it  very  hard  to 
get  a  good  concrete  at  and  below  that  point.  The  break  in  the 
form  work  is  also  objectionable  because  of  the  added  labor  and 

1  Journal  of  Western  Society  of  Engineers,  Vol.  20,  p.  653. 
*  P.  232,  et  seq. 


VARIOUS  TYPES  OF  WALLS 


139 


difficulty  of  pouring  the  concrete.  When  a  shape,  such  as  just 
shown  is  much  more  economical  than  the  straight  battered  back, 
it  will  be  found  that  the  counterforted  wall  will  prove  even  more 
economical,  and  should  therefore  be  adopted. 


FIG.  80. 


FIG.  81. 


Sloping  the  footing  as  shown  in  Fig.  81  may  prove  troublesome 
and  more  costly  in  the  end  than  the  plain  rectangular  section. 
Much,  of  course,  depends  upon  the  ability  of  the  contractor  to 
carry  out  the  niceties  of  the  design  and  it  is  thus  incumbent 
upon  the  engineer  planning  an  intricate  section  of  wall  to  see 
that  its  execution  is  placed  in  the  proper  hands. 

One  is  tempted,  in  designing  counterforted  walls  to  mold  cor- 
ners and  make  steel  details  as  shown  in  Fig.  82,  in  order  to  effect 
a  thorough  bond  between  the  slab  and  the  counterfort.  These 


FIG.  82. 

details,  again,  demand  extra  form  work,  steel  work  and  labor 
and  should  therefore  be  employed  with  due  appreciation  of 
the  possibility  of  their  added  expense. 

On  the  whole,  that  wall  is  most  effectively  and  economically 
designed  which  is  most  compactly  and  simply  shaped. 

jWith  the  rapid  development  of  thin  slab  construction  as 
markedly  shown  in  the  construction  of  concrete  ships  and  barges, 
there  is  excellent  promise  of  the  extension  of  such  work  to  re- 
taining walls.  If  the  construction  of  thin  slabs  and  intricate 


140 


RETAINING  WALLS 


details  becomes  commercially  applicable,  then  a  vast  field  is 
opened  to  economic  wall  design,  permitting  the  shape  to  follow 
every  peculiarity  of  the  environment  and  to  take  advantage  of 
whatever  economies  the  site  may  offer.  At  present  the  prac- 
tical limitations  of  construction  have  restricted  retaining  walls  to 
but  few  types  which  in  turn  are  limited  in  economic  thickness  by 
field  conditions. 

Problems 

1.  An  abutment  is  to  carry  two  tracks  as  shown  in  Fig.  83.  Each  of  the 
stringers,  under  full  load  brings  a  reaction  of  50  tons  upon  the  abutment. 
Determine  the  necessary  dimensions  of  both  a  gravity  and  a  reinforced 
concrete  "T"  wall. 

An  abutment  is  <a  combination  of  a  retaining  wall  and  a  pier.  Its  eco- 
nomical design  is  affected  not  only  by  the  type  adopted,  but  also  by  the  as- 
sumed location  of  the  girder  reaction.  In  the  case  of  a  gravity  wall,  the 


vertical  girder  reaction,  while  assisting  in  the  stability  of  the  wall,  may  by 
the  location  of  its  point  of  application,  induce  tensile  stresses  in  the  back  of 
the  wall.  Thus  in  Fig.  73,  the  girder  load  falls  within  the  outer  third, 
violating  an  essential  requirement  of  gravity  walls.  The  selection  of  a  type 
as  shown  in  Fig.  74  brings  the  girder  reaction  towards  the  center  of  the  wall 
and  assists  quite  materially  in  the  stability  moment  of  the  wall. 

The  distribution  of  these  girder  loads  may  be  assumed  to  follow  within 
planes  making  an  angle  of  30°  with  the  vertical  as  shown  in  Fig.  83.  The 
abutment  should  be  made  long  enough  to  permit  the  distribution  to  follow 
along  these  planes.  In  addition,  it  is  assumed  that  (for  reasons  given  in 
the  following  chapter)  the  abutment  is  independent  of  the  adjacent  struc- 
tures, so  that  the  span  loads  will  be  confined  within  the  abutment  proper 
as  shown  in  Fig.  83. 

Since  the  reaction  from  each  girder  is  100  kips,  the  area  for  bearing  upon 
the  concrete,  allowing  0.5  kip  per  square  inch,  is  200  sq.  in.  A  plate  12"  X 
18"  provides  this  bearing  area.  The  plate  will  be  placed  as  shown  in  Fig. 


VARIOUS  TYPES  OF  WALLS 


141 


84,  where  the  remaining  details  of  the  girder  seat  are  shown.  As  shown  in 
Fig.  83,  the  distribution  of  the  loads  spreads  between  a  distance  of  48', 
making  the  load  per  linear  foot  at  the  foot  of  the  abutment  40%g  =  8.3 
kips.  As  a  retaining  wall,  prior  to  the  setting  of  the  steel,  the  height  is 
30'  (above  the  footing)  without  any  surcharge.  From  Table  12,  a  face 
batter  of  5"  to  the  foot  will  give  the  necessary  dimensions  for  stability,  and 
will  also  satisfy  the  details  of  the  girder  seat. 

The  crane  load  is  taken  equivalent  to  500  pounds  per  square  foot.  The 
cases  are  lettered  and  discussed  in  the  same  order  as  on  page  129.  The 
graphical  analysis  is  shown  in  Fig.  85. 


FIG.  84. 


FIG.  85. — Graphical  analysis  of  abutment. 


(a)  The  resultant  intersects  the  third  point  (Checking  the  tabular  value) 
and  R  =  28.2  +  4.5  =  32.7 

Si  =  65/13.5  =  4.8  kips  per  square  foot.     The  permissible  soil  intensity 
in  this  and  the  following  work  is  taken  as  4  tons  per  square  foot. 

(6)  The  resultant  intersects  at  the  %7  point,  and  from  (39) 
66 


Si 


(2  —  3  X  0.185)  =  7  kips  per  square  foot;  which  is  within  the 


lo.O 

permissible  value. 

S2  =  T5~^  (  —  0.44)  =  2120    pounds    per  square  foot,  or  15  pounds  per 
lo.o 

square  inch.     This  tensile  stress  in  the  concrete,  developed  under  a  crane 
load  prior  to  the  setting  of  the  span,  is  a  permissible  stress. 

(c)  This  condition  is  quite  similar  to  the  preceding  one,  with  the  excep- 
tion that  the  indeterminate  factor  of  the  frictional  resistance  between  the 
girder  bearing  and  the  abutment,  together  with  the  dead  weight  of  the  span 
add  to  the  wall  stability. 

(d)  For  this  case  (that  of  full  loading)  the  resultant  is  found  to  intersect 
exactly  at  the  third  point.     R  =  42  kips 

$1  =  84/13.5  =  6.2  kips  per  square  foot. 

The  section,  then,  satisfies  all  the  necessary  conditions  of  design  and 
construction. 

Reinforced  concrete  section.  Assume,  as  in  the  case  of  the  ordinary  re- 
inforced concrete  retaining  wall,  the  criterion  of  economy,  i  =  e.  Let  the 
total  toe  pressure  not  exceed  7  kips  per  square  foot,  leaving  a  margin  for  the 


142 


RETAINING  WALLS 


toe  pressure  caused  by  the  girder  load.  Note  here,  that  since  a  skeleton 
section  of  wall  is  assumed,  with  the  point  of  application  of  the  resultant 
located  at  the  vertical  stem  of  the  wall,  the  girder  load,  which  is  at  the  same 
point,  can  have  no  effect  upon  the  wall  dimensions,  and  merely  increases 
the  intensity  of  the  soil  distribution.  From  (95),  with  Si  =  3.5  tons  per 
square  foot,  H  =  38'  (taking  the  thickness  of  footing  3  feet)  allowing  for  a 
five  foot  surcharge: 


e  =  6~6 


120  X  3.5 
38 


=  0.26 


Take  the  point  of  application  of  the  resultant,  and  the  location  of  the  face 
of  the  abutment  at  the  quarter  point  of  the  base.  From  Table  18  with 
this  value  of  e  and  i,  k  =  0.50  and  the  base  width  w  is,  accordingly  16.5 
feet.  With  a  girder  load  of  8.3  at  the  quarter  point,  from  (39) 

-3  (2  -  0.75)  =  1.25 


and  the  total  toe  pressure  is  8.25  kips,  a  permissible  excess  over  the  allowable 
4  tons  per  square  foot. 

The  height  of  the  vertical  stem  is  30',  and  from  Table  21  the  critical  height, 
above  which  the  shear  controls  the  thickness  of  the  stem  is  less  than  30'. 
The  thrust  for  the  given  surcharge  is  20  kips,  located  11.4  feet  above  the 
top  of  the  footing.  From.  (113),  the  thickness  of  wall  because  of  shear  is 

d  =  20/5.04  =  3.95 
A  thickness  of  4'  will  be  used  at  the  base. 

The  footing  moment  is  found  to  be  119  ft.  kips  and  the  depth  for  balanced 
reinforcement  is,  from  (101) 


requiring  a  thickness  of  3  feet.     If  no  special  stirrup  reinforcement  is  placed 
to  take  care  of  the  diagonal  tension,  an  excessive  depth  will  be  required  for 


FIG.  86. — Graphical  analysis  of  abutment. 

the  shear  (24.5  kips).  For  this  reason  it  will  be  assumed  that  such  rein- 
forcement is  employed  here  and  the  depth  of  the  slab  adopted  will  be  that 
required  by  the  bending  moment.  The  thickness  of  the  toe  extension  will 
also  be  taken  as  3  feet,  bearing  in  mind  that  the  thickness  of  the  footing, 


VARIOUS  TYPES  OF  WALLS 


143 


both  heel  and  toe,  must,  for  construction  reasons,  be  kept  the  same. 
The  introduction  of  concrete  fillets  at  the  junction  of  the  footing  and  arm 
would  obviate  the  need  for  web  rods  and  a  comparative  estimate  may  prove 
that  the  fillets,  with  the  extra  work  involved,  are  cheaper  than  the  compli- 
cated rod  details  of  web  reinforcement. 

Discussing  the  separate  cases  of  loading,  treated  graphically  in  Fig.  86, 
for  the  case  of  total  loading  (Case  d)  the  point  of  application  of  the  resultant 
is  at  e  =  4.75/16.5  =  0.288;  whence  from  (39),  with  R  =  59  kips,  Si  =  8100 
pounds  per  square  foot. 


0-4'  -->! 


FIG.  87. 

dround  Surface  ..-Surcharge  of  S:0" 


FIG.  88. 

Omitting  the  span  load  (Cases  b  and  c)  the  point  of  application  of  the 
resultant  is  at  e  =  4.5/16.5  =  0.273  and  with  R  =  51  Si  =  7.3  kips  per 
square  foot. 

The  section  as  shown  therefore  satisfies  the  governing  conditions.  The 
wall  should  be  recalculated,  using  the  dimensions  and  loadings  as  actually 
found. 

Fig.  87  shows  the  sections  of  the  gravity  and  reinforced  concrete  walls. 

2.  Find  the  stresses,  moments,  etc.,  in  a  box  section  as  shown  in  Fig.  88. 

It  is  necessary  to  make  a  preliminary  assumption  in  order  to  proceed  with 


144  RETAINING  WALLS 

the  analysis  of  this  section  under  the  theory  of  least  work.  For  this  reason, 
it  will  be  assumed,  tentatively,  that  the  moments  of  inertia  of  the  side-walls 
and  roof  are  equal.  Adding  two  feet  to  b  and  one  foot  to  h,  gives  the  dimen- 
sions along  the  gravity  axes  of  the  section.  The  value  of  e  is  now  2^6  = 
1.69.  From  Table  25,  Zi  =  0.23  and  Z2  =  2.38.  The  value  of  c  =  ^e 
=  0.875.  J  is  then  taken  at  its  usual  value  >£. 

For  roof  loading  alone 


For  side-wall  loading  alone 


1  v  1fi2 

H  =   3  X20   (3  +  6'38  X  2>38)  =  7'8  kips 

1  v  1ft* 
M  =  -  ~^-~  (1  -  4.38  +  6.38  X  2.38j  =  -  27.0  kips. 

For  simultaneous  loading 

H  =  7.8  -  3.7  =4.1  kips,  directed  outwards. 
M,  =  -27  +  20  =  -7  kip  feet. 

At  any  point  x,  above  the  base,  where  x  =  kh,  the  moment  is 
Mx  =  -7  +  Hkh  -  ^  [3(1  +  c  -  k)k2  +  2/c3] 
=  -  7  4-  66fc  -  22.7/c2(5.6  -  fc) 

For  the  various  values  of  k,  Mx  has  been  tabulated  as  shown  in  accompanying 
table.  The  roof  moment  at  any  point  y,  where  y  =  pb,  is,  taking  the  last 
found  value  of  Mx  as  given  in  the  table,  —  46, 

k  Mx 

0-7  M  =  -  46  +  510p(l  -  p) 

.1  —  2              A  table  has  been  similarly  prepared  for  a  set  of  values 

.2  +1           of  p,  up  to  the  center  of  the  span. 

.3  +2 

.4  0 

.5  -3 

.6  -8 

.7  -16 

.8  -24 

.9  -34 

1.0  -46 

p  M            The  assumption  that  the  roof  and  sidewalls  are  simul- 

0  —  46         taneously  loaded  does  not,  necessarily  give  the  maximum 

.  1  0         moments.     During  construction  it  is  quite  possible  that 

.  2  36         the  side  walls  will  be  loaded  up  to  the  roof  line,  before 

.3  61         any  load  is  placed  upon  the  roof.     The  only  roof  load 

.4  76         is  then  its  dead  weight,  which,  with  the  assumption 

.5  82        that  the  roof  is  two  feet  thick,  gives  a  load  of  0.3  kips 
per  foot.     There  is  a  triangular  distribution  of  pressure 

along  the  side  wall,  with  a  value  of  q  =  1600/3  =  0.53  kips. 


VARIOUS  TYPES  OF  WALLS  145 


For  roof  loaded  alone,  from  (160) 
H  =  '3  X,  27* 


- 

For  side  wall  loaded  alone,  from  (162) 

H  =  '^-~^  (3  +  2.38)  =  2.3 

M  =  -  '533  *  162  (1  +  4.76)  =  -13.1  kip  feet 
bO 

Under  the  simultaneous  loading 

H  =  1.5  directed  outwards. 
Mi  =  -  9kip  feet. 
As  before,  x  =  kh,  and  c  =  0 

Mx  =  -9  +  24fc  -  22.7fc2(3.6  -  /c) 
A  table  of  values  of  M  for  the  side  wall  is  given  here. 

k          M  The  roof  moment  is,-with  p  the  same  as  above, 

0-9  M  =  -  44  +  lllp(l  -  p) 

.1  —  7  A  table  of  these  moments  up  to  the  center  is  given  here. 
.2—7  p  M  A  further  condition  of  loading  may  be 

.3  -  8  0  -44  anticipated.  With  time  the  effect  of 
.4  —  1 1  .1  —  34  cohesion  may  materially  reduce  the  side- 
.5  — 15  .2  —  26  wall  pressure,  or  due  to  a  variety  of  con- 
.6  —19  .3  —21  ditions,  the  side  wall  pressure  may  be 
.7  -24  .4  -17  considerably  less  than  that  assumed. 
.8  -32  .5  -16  Let  this  state  of  loading  be  analyzed 
.9  -37  upon  the  assumption  of  a  full  roof  load- 

1.0       —  44  ing  and  a  sidewall  pressure  as  given  in 

the  work  immediately  preceding. 
For  roof  loading  alone,  from  before 

H  =  3.7;    M  =  19.7  ft.  kips 
For  the  side  wall  loading  as  assumed 

H  =  2.3  and  M  =  -13.1  ft.  kips 

The  net  thrust  due  to  both  loadings  is  1.4  directed  outwards,  and  the  mo- 
ment is  +6.6  ft.  kips. 

Mx  =  6.6  -  22fc  -  22.7/c2(3.6  -  k) 

The  tabular  values  for  the  moments  in  the  sidewall  are  again  shown  in  the 
accompanying  table. 

k          M  The  roof  moment  is 

0         +7  -74  +  510p(l  -  p) 

.1       +4  The  values  for  this  moment  up  to  the  center  of  the 

.2       +1        span  are  given  in  the  table. 
.3-7         p  M 

A       -14          0         -74 
.5       -22         .1         -28 
.6       -31          .2         +8 
.7       -40         .3         +33 
.8       -52         .4         +48 
.9       -63         .5         +53 
1.0       -74 
10 


146 


RETAINING  WALLS 


The  structure  is  designed  to  satisfy  the  maximum  moments  shown  in  the 
diagrams.  The  maximum  roof  moment  is  82  with  practically  an  equal  but 
opposite  moment  at  the  fixed  corner.  The  thickness  for  balanced  reinforce- 
ment is  found  to  be  2.25  feet.  The  steel  ratio  0.0075,  requires  2.4  square 
inches  per  linear  foot;  too  heavy  a  reinforcement.  A  thickness  of  33",  or 
3  feet  overall  is  finally  adopted,  which  requires  a  steel  reinforcement  of  1 
inch  square  bars  spaced  6".  The  maximum  side  wall  moment  will  occur 
about  at  k  =  0.9  (since  the  roof  is  3'  thick),  whence  M  =  -63  ft.  kips. 
'Again,  although  balanced  reinforcement  needs  a  2'  slab,  to  keep  the  rod 
weight  within  reasonable  limits  a  27"  slab  will  be  used,  with  an  overall 
dimension  of  2'  6".  For  this  condition  1"  bars  6"  apart  are  required. 


~T~ 

:  0 
-.I 


6-0- -: 


Fftods  Id  "C.toC.  between  these  Points 


twe'e 


l"aRods,6"C.foC 


v 


FIG.  89. 

The  moments  of  inertia  of  these  sections,  it  is  noticed,  do  not  fulfill  the 
assumed  condition.  To  take  the  ratio  as  found  for  the  sections  above,  will 
again  prove  slightly  incorrect  in  the  final  analysis,  and  for  this  reason  an 
intermediate  value  of  the  moment  of  inertia  ratio,  between  that  first  assumed 
and  that  now  found  will  be  used.  The  moments  of  inertia  of  rectangular 
sections,  of  the  same  width  are  to  each  other  as  the  cubes  of  their  depths. 
The  ratio  72//i  =  15.6/27  =  0.58.  The  average  of  this  value  and  the  value 
1,  first  taken  is  0.79.  The  value  of  e  is  now  1.3,  making  Zi  and  Z2  0.28  and 
2.35  respectively. 

In  tabular  form  the  moments  at  the  three  important  points,  for  the  three 
conditions  discussed  above  are 


CONDITION  OF  LOADING 
C 

Full  roof  and  sidewall —  2 

Dead  weight  roof  and  light  wall —  5 

Full  roof  and  light  wall +11 


A  Center  of  roof 
-56  +71 

-50  -25 

-83  +44 


VARIOUS  TYPES  OF  WALLS  147 

It  is  seen  that  quite  a  large  variation  in  the  assumed  values  of  the  moment 
of  inertia  ratio  has  but  sluggish  effect  upon  the  moments  and  it  is  probably 
safe  to  take  both  the  roof  and  sidewalls  of  the  same  thickness,  subject  to  a 
bending  moment  of  70  foot  kips  at  the  center  of  the  roof  and  at  the  upper 
fixed  corners,  and  to  a  negative  moment  of  —  25  foot  kips  at  the  center  of 
the  roof. 

The  final  section  must  take  care  of  the  moments  throughout  the  frame 
detailed  in  accordance  with  the  adhesion  requirements  and  bent  in  accord- 
ance with  the  bearing  formulas  given  in  the  preceding  chapter.  Fig.  89 
gives  a  layout  of  the  section,  with  the  rod  layouts  as  indicated  by  the 
previous  work. 

It  must  again  be  emphasized  that  the  stresses  existing  in  a  structure  of 
this  character  are  quite  different  from  those  which  are  found  upon  analyzing 
the  structure  into  its  separate  members  and  when  a  subsurface  structure  is 
built  as  shown  above,  provision  must  be  made  for  the  distribution  of  stresses 
as  given  by  the  analysis  just  made. 

The  Selection  of  an  Economical  Type.1 — While,  clearly,  for 
some  given  height,  a  counterforted  wall  becomes  cheaper  than  a 
cantilever  wall,  a  search  of  pertinent  literature  fails  to  yield  any 
method  of  obtaining  such  a  height,  save  by  actual  comparison  of 
two  completed  designs.  It  may  be  well  worth  while  to  establish 
some  method  of  obtaining  this  "  critical"  height. 

It  is  true,  extraneous  factors  may  control  the  selection  of  types 
of  walls  and  the  dimensions  of  the  component  members,  but 
generally,  a  wall  is  so  designed  as  to  satisfy,  most  economically, 
its  stresses. 

Again,  the  bending  moment,  shear,  or  bond  stress,  may  each 
in  turn  control  the  necessary  thickness  of  the  several  parts  of  the 
wall,  as  the  height  is  varied.  It  is  to  be  noted  that,  with  few  ex- 
ceptions, such  several  stresses  usually  require  about  the  same 
thickness  of  section,  though  probably,  a  greater  variation  in  the 
amount  of  reinforcement  required.  In  assuming  that  the  wall 
dimensions  follow  the  theoretical  requirements  a  large  percentage 
of  actual  cases  are  covered  and,  if,  further,  these  dimensions  are 
taken  in  accordance  with  the  stress  of  simplest  expression,  no 
serious  error  results.  With  this  in  mind,  the  various  thicknesses 
of  both  the  cantilever  and  the  counterforted  walls  are  those 
selected  in  accordance  with  the  bending-moment  requirements. 

In  the  work  that  follows,  since  it  is  a  comparative  estimate  of 
the  cost  of  the  two  types  that  is  sought,  it  is  justifiable  to  select 
as  a  type  for  the  present  analysis,  that  involving  the  least  mathe- 
matical analysis.  It  is  quite  clear  that  variations  in  the  toe 

1  Reprinted  from  Engineering  and  Contracting,  Feb.  26,  1919. 


148  RETAINING  WALLS 

length  or  in  the  assumed  position  of  the  resultant,  will  not  affect, 
to  any  material  extent,  the  comparative  estimate.  For  this 
reason,  the  condition  for  economy  as  given  on  page  82  is  adopted 
here,  with  a  further  provision,  that  e  =  J«j,  the  usual  soil  pres- 
sure distribution.  With  these  conditions  (91)  then  becomes 


_i  IT 

2VT 


k       A     /A  -r  3c 


The  dimensions  for  the  "T"  cantilever  are  taken  as  follows: 
the  thickness  of  the  base  of  the  vertical  arm,  from  (112)  is 

dv  =  0.0185 

and  the  thickness  of  the  top  of  the  arm  is  taken  at  its  usual  mini- 
mum value  one  foot.  For  the  footing,  from  (119)  I  is  about  0.7 
and  the  required  thickness  of  the  footing  slab  is  then  \/.7  or  0.84 
times  the  arm  base  thickness.  For  the  counterfort  wall,  from 
(126)  with  the  usual  value  of  the  constants  the  thickness  of  the 
vertical  slab  is 


d'v  =  0.0132m  \h(  1  +  c)  =  C', 
and  that  of  the  footing,  from  (138)  is 


The  counterfort  itself  is  usually  one  foot  thick  and  will  be  so  taken 
here. 

The  cost  of  the  steel  rods  is  a  small  part  of  the  total  cost  of  the 
wall  and  the  relative  difference  of  the  cost  of  the  steel  rods  in  the 
two  types  of  walls  would  thus  be  negligible. 

The  amount  of  face  and  rear  forms  for  the  vertical  arm  of  both 
types  is  substantially  the  same  and  will  not  enter  into  the  com- 
parative estimate.  The  variable  factors  in  the  comparative 
estimate  are  then  :  the  amount  of  concrete  in  either  type  and  the 
forms  required  for  the  counterfort  itself. 

Let  L  be  the  total  length  of  wall  under  consideration,  r  be 
the  cost  of  placing  concrete  into  the  forms  (the  cost  is  practically 
the  same  for  both  types)  and  let  t  be  the  cost  of  the  form  work  and 
necessary  bracing,  per  square  foot  of  concrete  face  supported. 
For  the  counterforted  wall  the  amount  of  concrete  is 

L(d'vh  +  khd'j  +  ~  —* 


VARIOUS  TYPES  OF  WALLS  149 

and  its  total  cost 

Lrhld'9(l  +  fc\/3)  +  TT- 
Zm 

\  * 

The  cost  of  the  face  forms  for  the  counterfort  is 

tm~2'2 

making  the  total  variable  cost  of  the  counterfort  wall 

Lrh\d'v(l  +  &V3)  +  <^(1  +2^)  }  (167) 

The  volume  of  the  "  T"  cantilever  is 

L  ( n~^~h  +  khdb }  =  Lh   ~  +  dv  (^  +  0.84A;j 

and  its  total  cost 

Lhrl^  +  dv  (i  +  0.84fc)  1  (168) 

Equating  (167)  and  (168) 

d'v(l  +  /c\/3)  +  ^(l  +  2-*)    =  0.5  +  ^(0.5  +  0.84A;) 


Replacing  the  thicknesses  of  the  sections  by  their  values  given 
above 


(0.5  +  0.84/b)     (169) 

Later  it  will  be  shown  that  the  economic  spacing  of  the  counter- 
forts is  given  by 

m  =  3.1  Rhy* 
where 

R--V/1  +  2-J 

With  this  value  (169)  becomes 

CJi**  -  RCihX  +  M  =  0 
a  quadratic  in  h^ 

k 


with  C2  =  .0132  V1  +  c    3.1  (1-f  A;V3)  + 
and  C"i  =  .0186\/1  +  3c  (0.5  +  0.84k) 


mu  1  *  LV  •     ^C'l  +  V#2Ci2  -  2C 

The  value  of  ft«  is  . 


150 


RETAINING  WALLS 


Table  26  gives  a  series  of  values  of  this  critical  height  h  for  several 
values  of  the  cost  ratio  t/r  and  the  surcharge  ratio  c. 

TABLE  26 


\'/r 

K 

M 

K 

i 

C   \ 

0 

15 

22 

28 

33 

M 

11 

17 

22 

27 

M 

10 

15 

19 

23 

Economic  Spacing  of  Counterfort.— To  determine  the  spacing 
of  the  counterforts  to  give  the  most  economic  wall  sections,  it  is 
seen  that  (167)  is  the  required  expression  for  the  variable  cost  of 
the  counterforted  wall  as  the  spacing  of  the  counterforts  change. 
If,  by  the  theory  of  Maxima  and  Minima,  the  derivative  of  this 
expression  with  respect  to  m,  is  put  equal  to  zero,  there  results, 
after  replacing  the  several  thicknesses  by  their  values  as  previously 
found 

h20) 


!".(!  +  fc-v/3) 


£ 


k 


V/.0132\/l-f  c(l+A;\/3) 
With  R  as  given  above,  and  noting  that  the  expression 


(i  +  fcyi  +  c 

after  using  the  value  of  k  as  given  in  (91)  is  practically  constant 
and  equal  to  J£,  this  expression  becomes 


m  = 


m 

Table  27  gives  a  series'of  values  of  m  for  the  several  values  of 
t/r  and  the  height. 

TABLE  27 


H 

15 

20 

25 

30 

35 

40 

50 

H 

7.5 

8.1 

8.6 

8.9 

9.3 

9.6 

10.2 

% 

8.6 

9.3 

9.8 

10.2 

10.7 

11.0 

11.6 

H 

9.6 

10.4 

11.0 

11.5 

12.0 

12.4 

13.1 

i 

10.6 

11.4 

12.0 

12.6 

13.1 

13.5 

14.3 

It  is  reasonable  to  expect  that  the  laws  governing  the  theory  of 
probabilities  hold  here  and  that,  ^therefore,  the  small  errors 
introduced  in  the  above  approximations  are  fairly  compensatory. 


PLATE  II 


11 


« 


PLATE  III 


FIG.  C. — Crack  at  sharp  corner  of  wall  due  to  tension  component  of  thrust. 


CHAPTER  V 

TEMPERATURE  AND  SHRINKAGE  STRESSES,  EXPANSION  JOINTS, 

WALL  FAILURES 

In  the  setting  and  curing  of  concrete  and  in  the  seasonal  varia- 
tions in  temperature,  stresses  are  induced  in  retaining  walls 
which,  because  of  the  longitudinal  continuity  of  the  wall,  must 
be  resisted  by  the  material  itself.  Plain  concrete  monoliths,  un- 
reinforced,  will  crack  at  well  defined  intervals  because  of  failure 
of  the  material  through  tension.  It  is  quite  difficult,  despite 
the  insertion  of  rods  to  prevent  cracks.  It  is  possible,  however, 
by  properly  introducing  rods,  to  concentrate  the  tendency  to 
cracking  at  assigned  intervals  and  then,  to  avoid  unsightly 
breaks,  to  place  an  actual  joint  at  such  places.  Reinforced 
walls  are  at  times  built  without  any  joints  and  seem  to  have 
such  proper  reinforcement  that  no  cracks  are-  apparent. 

A  theoretical  discussion  of  the  temperature  changes  that  may 
be  expected  within  masonry  masses  may  be  interesting  as  indicat- 
ing the  expected  amount  of  stresses  to  be  anticipated  by  rod 
reinforcements. 

It  is  patent,  that  the  further  from  the  exposed  surface  a  point 
is  within  the  mass,  the  smaller  will  be  the  variation  of  tempera- 
ture at  that  point  for  any  given  surface  range  of  temperature. 
Experiments  have  been  made  to  determine  this  range  at  various 
points,  covering  quite  long  periods  of  time1  and  in  recent  masonry 
dam  construction,  automatic  temperature  recording  devices 
have  been  incorporated  in  the  work  so  that  an  exhaustive  record 
of  the  variation  of  temperature  is  available. 

It  seems  desirable  to  attempt  to  express,  mathematically,  this 
distribution  of  temperature  and,  in  view  of  the  fact  that  the 
theoretical  results  so  obtained  are  reasonably  in  accord  with  the 
experimental  results,  they  should  prove  of  service  in  making 
provision  for  temperature  stresses  in  masonry  structures. 

1  Trans.  A.S.C.E.,  Vol.  Ixxix,  p.  1226. 

151 


152  RETAINING  WALLS 

The  variation  of  seasonal  temperatures  at  the  surface  may  be 
given  by  an  expression  of  the  form, 

u*A+Bco8~JFi  (170) 

in  which  u  is  the  temperature,  A  and  B  are  constants,  T  is  the 
period  of  change  and  I  is  the  time. 

In  the  distribution  of  heat  through  large  masses,  where  the 
temperature  at  the  surface  is  a  function  of  the  time,  it  can  be 
shown1  that  the  temperature  u  at  any  distance  x  from  the  surface 
at  the  time  I  is 

u  =  A  +  Be~kxcos  (2ir/T  -  kx)  (171) 

in  which  e  is  the  base  of  natural  logarithms  and  k  =  --\fe  «2  is 

known  as  the  coefficient  of  thermal  diffusivity,  which,  for  concrete 
(Smithsonian  Physical  Tables)  is  0.0058  in  the  C.G.S.  system. 
The  maximum  range  of  temperature  occurs  between  t  equal 
any  integer  say  n  and  t  =  n  +  J^.  At  the  surface  this  range 
becomes,  from  (170)  2B;  at  any  point  x  from  the  surface  the 
range  is  from  (171)  2Be~kx  cos  kx.  The  ratio  of  the  range  at  any 
point  x  to  that  at  the  surface  is 

e~kx  cos  kx  =  Ix  (172) 

and  if  U  is  the  surface  range,  that  at  any  plane  x  away  from  the 
surface  is  UIX. 

In  discussing  seasonal  changes,  the  period  T  is  one  year,  which 
must  be  expressed  in  seconds  in  accordance  with  the  diffusivity 
constant  a2.  For  this  period,  and  for  concrete  k  =  0.00413. 
Table  28  shows  a  comparison  with  the  results  from  the  formula 
and  those  experimentally  found  in  the  records  quoted  above.2 

The  daily  range  may  in  itself  be  taken  as  periodic  and  expressed 
by  (170)  and  (171).  For  this  period,  one  day  expressed  in  sec- 
onds k  =  0.079.  Table  29  gives  a  parallel  comparison  between 
the  theoretical  and  the  experimentally  determined  range. 

It  is  seen,  from  a  study  of  the  daily  variation  of  temperatures 
that  the  surface  range  is  rapidly  decreased  a  few  inches  from 
the  surface.  In  designing  masonry  structures  it  is  sufficient,  in 
making  provision  for  the  temperature  range  to  take  a  seasonal 
range  based  on  about  weekly  averages.  For  climates  in  the 

1W.  E.  BYERLY,  "Fouriers  Series  and  Spherical  Harmonics,"  p.  89. 
2  Tables  for  ,~x  are  to  be  found  in  PIERCE.  "A  Short  Table  of  Integrals." 


TEMPERATURE  AND  SHRINKAGE  STRESSES      153 


Middle  Atlantic  States,  this  range  is  about  40°  either  way  from 
the  mean. 


TABLE  28 


TABLE  29 


X 

/ 

Theoretical 
range 

Actual 
range 

| 

0.0 

1.00 

75            75 

1.0 

.87 

65 

2.0 

.76 

57 

3.5 

.57 

43 

32 

5.0 

.42 

31 

10.0 

.09 

7         i     12 

20.0 

.04 

3 

0 

• 

X 

Ix 

Theoretical 
range 

Actual 
range 

o. 

1.00 

50 

50 

.25 

JC 
.  ^rO 

22 

.50 

.11 

5 

1.0 

.07 

3 

2 

1.5 

.02 

1 

2.0 

.01              1         j       1 

2.5 

.002           0 

3.0 

.000 

3.5 

0 

1 

If  the  unit  stress  developed  by  a  change  of  one  degree  in  the 
temperature  is  s  and  if  the  surface  range  is  U,  then  the  stress  at 
any  x  is  sUIx  and  the  total  stress  across  a  section  of  thickness 
w  and  unit  width  is 


where 


and  the  average  unit  stress  over  the  section  is  csU.     Table  30 
gives  the  value  of  c  for  various  values  of  w. 

TABLE  30 


sUj]w!xdx  =  sUj^we~kxcos  kxdx 

=   sUcw,  (173) 

cw  =  n  {  e~*w(sin  kw  —  cos  kw)  +  1  1  (174) 


Seasonal  change 


w 

c 

j 

1 

.95 

.48 

2 

.87 

.47 

3 

.82 

.46 

4 

.75 

.43 

5 

.70 

.42 

6 

.65 

.41 

7 

.60 

.39 

8 

.55 

.37 

9 

.51 

.35 

10 

.47 

.33 

154  RETAINING  WALLS 

If  E  denotes  the  modulus  of  elasticity  for  masonry  and  n  the 
coefficient  of  expansion, 

s  =  nE  (175) 

For  concrete  this  value  of  s  is  about  ten  pounds  per  square  inch, 
for  every  degree  change  in  temperature  (Fahrenheit) . 

Replacing  w  in  (173)  by  the  area  of  the  concrete  section  Ae, 
the  total  stress  across  a  section  is 

csUAc.  (176) 

Let  the  range  of  temperature  where  the  steel  rod  is  to  be  placed 
be  V  and  let  the  area  of  steel  be  A8,  with  the  ratio  of  steel  to 
concrete  area,  as  before  p.  The  stress  developed  in  the  steel  by  a 
change  of  one  degree  is  s'  and  will  be  ns}  with  n  the  ratio  of  the 
two  moduli  (see  page  86).  The  total  stress  across  a  section 
because  of  a  surface  range  of  U  is  then 

csUAc  +  Ass'U'.  (177) 

The  concrete  can  take  fe  pounds  per  square  inch  before  failure 
and  the  steel  can  take  fa  pounds  per  square  inch  up  to  its  elastic 
limit.  The  resisting  section  to  the  above  temperature  stress 
is  thus 

f8As  +  feAc  =  fspAc  +  fcAc  (178) 

Equating  (177)  and  (178)  and  solving  for  p 

.  csU  -  A  M79) 

p  -/.  -s'w  (    } 

For  example,  take  a  range  from  the  mean,  as  above  of  40°,  and 
average  slab  thickness  of  two  feet,  fe  =  200  pounds,  and  fs  = 
45,000  pounds.  From  the  Table  30  c  =  0.87,  and  since  for  a 
cantilever  wall,  where  the  vertical  rods  are  at  the  rear  face  it  is 
customary  to  likewise  place  the  check  rods  (for  convenience  of 
construction)  at  the  rear  face  from  Table  28  Ix  =  0.76,  whence 
U'  =  0.76  X  40°  =  30°.  The  required  ratio  of  steel  is  then,  from 
(179)  with  sf  =  15  X  11  =  165 

0.87  X  10  X  40  -  200 
P         45,000  -  165  X  30° 

Specifications  usually  require  about  %  of  one  per  cent,  of  steel 
for  temperature  reinforcement,  which  agrees  fairly  well  with  the 
above  value  just  found.  It  is  seen  that  a  steel  of  high  elastic 


TEMPERATURE  AND  SHRINKAGE  STRESSES      155 

limit  should  be  specified.  The  expansion  coefficients  of  both 
steel  and  concrete  are  fairly  alike  so  that  there  is  no  stress  in- 
duced between  steel  and  concrete  because  of  this  temperature 
change. 

Shrinkage.  —  -Unlike  temperature  stresses,  the  stress  due  to 
shrinkage  is  induced  in  the  steel  by  the  action  of  the  concrete 
in  curing  and  drying  out.  While  there  is  little  definite  regarding 
the  theory  of  shrinkage  experimental  data  has  shown1  that  the 
shrinkage  of  concrete  is  about  0.0004  of  the  length.  In  the  same 
paper  the  stress  due  to  the  shrinkage  is  given  by  the  expression 

-          (180) 


•          - 

C  is  the  coefficient  of  shrinkage  (given  above)  E  the  concrete 
modulus,  n  and  p  the  usual  concrete  functions.  The  stress 
induced  in  the  steel  is  then 

/.  =  f./P  (181) 

With  the  amount  of  reinforcement  as  specified  for  tempera- 
ture stresses,  the  concrete  stress  is  seen  to  be,  from  (181)  40 
pounds  per  square  inch  and  the  corresponding  steel  stress  about 
12,000  pounds  per  square  inch. 

To  provide  for  temperature  and  shrinkage  stresses  the  rods 
should  be  placed  at  right  angles  to  those  put  into  take  care  of  the 
earth  pressure  stresses.  Since  the  maximum  temperature  ranges 
occur  at  the  surface,  it  is  desirable  but  not  necessary  that  the 
rods  be  placed  at  the  surface.  It  has  been  seen  that  for  the  canti- 
lever walls  it  is  not  feasible  to  place  the  rods  at  the  face.  Gener- 
ally these  rods  are  woven  in  with  the  vertical  stress  rods. 

Settlement.  —  The  settlement  of  a  wall  is  intimately  connected 
with  the  character  of  its  foundation.  From  the  discussion  on 
foundations  in  Chapter  2,  it  was  seen  that  certain  types  of  soil 
require  a  distinct  distribution  of  loading;  the  more  yielding  the 
soil  was,  the  more  urgent  it  became  that  the  distribution  of  soil 
pressure  be  a  uniform  one.  It  is  generally  agreed,  that,  within 
reasonable  limits  (these  limits  determined  by  the  structures 
adjacent  to  or  supported  by  the  wall)  a  uniform  settlement  of 
the  wall  is  harmless,  since,  with  a  proper  spacing  of  expansion 
joints,  or  with  carefully  distributed  reinforcement,  no  cracking 
will  occur  in  the  wall  body.  Unequal  settlement  produces 

1  See  Bulletin  No.  30,  Iowa  State  Agricultural  College. 


156 


RETAINING  WALLS 


cracks,  which  not  only  prove  unsightly,  but  may  indicate  incipi- 
ent failure. 

Unequal  settlement  may  be  expected  on  yielding  soils  where 
the  distribution  of  pressure  is  not  a  uniform  one;  where  the  char- 
acter of  the  soil  changes,  one  type  yielding  more  than  the  other 
type;  at  junctions  of  new  and  old  work,  the  old  work  having 
settled  with  the  soil,  the  new,  in  gradually  taking  up  its  settle- 


Deformect 
Bars  " 
"Railroad  Rails 


FIG.  90. — Bottoms  reinforced  because  of  threatened  settlement. 

ment,  necessarily  destroying  the  bond  between  the  new  and  old 
work.  The  remedies  for  these  are  quite  obvious.  For  the  first 
case  it  has  been  sufficiently  emphasized  that  there  must  be  a  uni- 
form distribution  of  pressure.  A  joint  should  be  placed  in  the 
wall  wherever  the  character  of  the  soil  changes  and  especially 
between  a  yielding  and  non-yielding  soil.  Joints  should  also  be 
placed  between  new  and  old  work.  It  is  a  good  detail,  where 


„  Rods  in  Vertical 
Arm 


FIG.  91. 

settlement  is  expected,  to  reinforce  the  bottom  of  the  footing 
with  longitudinal  rails  or  rods  as  shown  in  Fig.  90.  Such  rein- 
forcement will  tend  to  distribute  any  impending  movement  and 
thus  prevent  a  crack. 

While  of  common  occurrence  it  is  poor  practice  to  make  a 
wing  wall  monolithic  with  the  abutment,  save  on  unyielding  soils. 
The  character  of  loading  for  each  type  is  radically  different  mak- 


TEMPERATURE  AND  SHRINKAGE  STRESSES     157 

ing  unequal  settlement  inevitable.  Reinforcement  across  the 
junction  of  the  two  walls  is  uncertain  and  cracking  may  occur 
despite  such  rods.  A  photograph  (Plate  No.  2a)  and  Fig.  91  are 
given  illustrative  of  this. 

While  settlement  is  an  uncertain  problem,  careful  attention  to 
the  foregoing  points  will  reduce  to  a  minimum  the  chances  of 
cracks  on  these  accounts.  Where  the  face  of  the  wall  is  to  re- 
ceive special  treatment  or  is  to  be  panelled,  it  is  vital  that  every 
precaution  be  taken  against  unsightly  cracks.  As  in  the  case 
of  foundations,  the  provisions  to  be  made  against  expected  set- 
tlement demand  most  mature  engineering  judgment.  A  large 
crack  in  a  wall  is  usually  an  indication  of  lack  of  engineering 
foresight  and  where  such  work  is  adjacent  to  public  highways, 
becomes  unpardonable. 

Expansion  Joints. — -Where  movement  is  expected  in  a  wall,  due 
to  any  of  the  interior  or  exterior  changes  discussed  in  the  fore- 
going pages,  it  is  customary  to  attempt  to  localize  such  movement 
to  small  sections  of  the  wall.  For  this  purpose,  vertical  joints 
are  placed  in  the  wall  at  regular  intervals  and  are  constructed  so 
that  no  movement  can  be  carried  vertically  or  longitudinally 
across  them.  Since  it  is  desirable  that  a  wall  be  kept  in 
good  line,  the  joints  are  usually  so  built  to  prevent  transverse 
movement. 

In  a  monolithic  gravity  wall,  joints  are  essential  and  are  cus- 
tomarily spaced  at  from  30  to  50  feet  intervals.  This  makes 
ample  provision  for  temperature  and  shrinkage  stresses  and  makes 
it  possible  to  have  complete  concrete  pours  from  joint  to  joint. 
An  excellent  detail  of  such  a  joint  is  shown  in  Fig.  92,  giving 

—-OneSection — ->j 


tr 


Coat  of  Pitch 

FIG.  92. — Expansion  joints. 

freedom  of  movement  in  every  direction  except  a  transverse  one. 
One  section  of  wall  is  poured  completely  between  the  joints. 
After  the  joints  are  given  a  coat  of  some  tar  or  asphalt  prepara- 
tion the  adjoining  sections  are  then  poured.  To  prevent  seepage 
of  water  into  the  joint,  several  layers  of  fabric  and  tar  are  placed 
over  the  back  of  the  joint  and  extend  about  1J^  feet  on  either  side 
of  it  and  from  the  row  of  weep  holes  at  the  bottom  of  the  wall  up 
to  the  top  of  the  wall. 


158  RETAINING  WALLS 

While,  theoretically,  steel-concrete  walls  can  so  be  reinforced 
that  expansion  joints  are  unnecessary,  such  implicit  confidence 
in  the  theoretical  action  of  such  rods  is  not  wholly  warranted 
and  expansion  joints  are  usually  placed  with  about  the  same 
frequency  as  in  plain  concrete  walls.  The  check  rod  system  then 
distributes  all  movement  to  these  joints  and  the  wall  is  surely 
safe  against  cracking.  Mr.  Gustav  Lindenthal1  has  stated  that 
expansion  joints  are  a  source  of  danger  because  of  the  possible 
accumulation  of  water  in  them  with  a  threatened  wedge  action 
due  to  ice  formation.  Accordingly,  in  the  walls  of  the  New 
York  Connecting  Railroad,  described  on  page  127,  no  joints  were 
used,  full  dependence  having  been  placed  in  J^  per  cent,  of  rein- 
forcement to  take  up  whatever  secondary  stresses  were  induced 
by  temperature  changes,  shrinkage  and  settlement.  General 
engineering  practice  is,  however,  not  in  accord  with  this  view 
and  expansion  joints  are  almost  universally  used  in  reinforced 
concrete  walls. 

The  details  of  an  expansion  joint  for  the  cantilever  wall  are 
simple  and  may  be  made  the  same  as  the  detail  for  the  gravity 
wall  shown  in  Fig.  92.  For  the  counterforted  and  other  slab 
types  of  wall,  a  break  cannot  be  made  in  the  face  without  provid- 
ing a  special  detail.  It  is,  of  course,  possible,  in  the  case  of 
counterforted  walls,  to  build  two  adjoining  counterforts  with  the 


f Expansion  Joint  /  Cantilever  Arms 


':Rods  to  take          \ 
Cantilever  Moments' 

FIG.  93.  FIG.  94. 

joint  immediately  between  them  as  shown  in  Fig.  93,  but  such  a 
detail  is  necessarily  a  costly  one  and  to  be  avoided.  Generally 
the  joint  is  made  midway  between  the  two  buttresses  and  the 
slab  in  between  is  made  up  of  two  cantilevers  as  shown  in  Fig.  94. 
The  bottom  slab,  buried  in  the  ground  can  usually  be  made  con- 
tinuous and  the  expansion  joint  need  only  extend  to  the  bottom 
of  the  vertical  slab.  This  applies  equally  well  to  the  cantilever 
type  of  wall. 

In  stone  masonry  walls  it  is  inexpedient  to  place  any  joints  in 
the  wall,  but  where  the  stones  have  carefully  been  bedded  any 

1  Engineering  News,  Vol.  73,  p.  886. 


TEMPERATURE  AND  SHRINKAGE  STRESSES     159 

movement  is  usually  taken  up  and  distributed  by  the  mortar 
joints.  It  is  essential,  of  course,  that  there  be  the  proper 
ratio  of  headers  to  stretchers  to  effectively  distribute  all  such 
movements. 

Construction  Joints. — Any  break  in  the  continuity  of  pouring 
a  wall,  other  than  at  an  expansion  joint,  leaves  a  joint  in  a  wall, 
which  is  usually  termed  a  construction  joint.  It  is  not  generally 
possible  to  pour  a  section  of  a  wall  between  expansion  joints 
completely  in  one  continuous  operation.  It  is  impractical, 
usually,  to,  indicate  such  construction  joints  in  advance,  due  to 
the  exigencies  of  field  conditions.  The  steps  in  pouring  are 
generally :  the  bottom  slab  is  poured ;  the  vertical  is  later  poured 
in  as  few  operations  as  possible.  While  such  a  sequence  does  not 
give  the  ideal  location  for  such  joints,  by  the  proper  keying  and 
cleaning  of  the  construction  joints,  the  strength  of  a  wall  may  be 
satisfactorily  maintained.  It  may  be  interesting  to  note  a  series 
of  tests  on  the  efficiency  of  various  modes  of  treating  a  construc- 
tion joint  to  insure  a  proper  bond  between  the  old  and  new  work. 

H.  St.  G.  Robinson,  Minutes  of  the  Proceedings,  Inst.  of  C.  E., 
Vol.  clxxxix,  1911-1912,  Part  III,  p.  313,  has  performed  the  fol- 
lowing series  of  tensile  tests  taking  the  efficiency  of  a  solid  prism 
as  100  per  cent.  A  series  of  five  tests  upon  this  solid  prism  gave 
an  average  ultimate  strength,  in  tension,  of  329  pounds  per  square 
inch. 

For  the  abutting  faces  (new  and  old)  merely  wetted,  the  effi- 
ciency of  such  a  joint  was  38.3  per  cent,  of  the  solid.  A  series 
of  five  tests  gave  an  average  ultimate  strength  of  the  joint  of  126 
pounds  per  square  inch. 

For  the  abutting  faces  roughened  and  wetted  the  efficiency  was 
56.2  per  cent,  of  the  solid.  A  series  of  six  tests  gave  an  average 
ultimate  strength  of  the  joint  of  185  pounds  per  square  inch. 

For  the  abutting  faces  treated  with  acid  the  efficiency  of  the 
joint  was  82  per  cent,  of  the  solid.  An  average  of  six  tests  gave 
an  ultimate  strength  of  270  pounds  per  square  inch. 

For  the  abutting  faces  roughened  and  grouted  the  efficiency 
of  the  joint  was  85.5  per  cent,  of  the  solid.  An  average  of  four 
tests  gave  an  ultimate  strength  of  the  joint  of  281  pounds  per 
square  inch. 

From  the  above  it  is  evident,  that  by  cleaning  and  grouting  the 
surface  on  which  the  new  concrete  is  to  rest  almost  the  full  effi- 
ciency of  the  joint  will  be  attained. 


160 


RETAINING  WALLS 


It  must  be  noted  that  construction  joints  in  the  face  of  a  wall 
leave  a  permanent,  and  often  unsightly  mark.  This  matter  is 
discussed  somewhat  in  detail  in  a  later  chapter. 

It  is  now  possible  to  complete  the  reinforced  concrete  design  of 
Chapter  3.  The  secondary  rod  system  for  temperature,  shrink- 
age and  settlement  may  now  be  added  to  the  sections  shown 
in  that  chapter.  For  simplicity  of  construction  the  rods  are 
usually  attached  to  the  primary  system  of  the  wall.  In  the 
"L"  and  "T"  walls  the  rods  are  horizontal  as  shown  in  Fig.  95. 
If  the  distance  between  expansion  joints  is  too  large,  or  if  there 
are  no  expansion  joints,  it  becomes  necessary  to  splice  these  rods. 
The  rods  are  carried  beyond  the  point  of  splice  each  a  distance 
sufficient  to  develop  the  rod  in  adhesion. 


.Check  Rods 


Check  Rods. 


FIG.  95. 


FIG.  96. 


While  strictly,  such  rods  are  unnecessary  in  the  footing,  they 
will  act  as  a  distributing  system  in  case  of  threatened 
settlement. 

For  the  counterfort  and  other  slab  sectioned  walls,  the  check 
rods  are  vertical  and  placed  at  the  outer  face,  see  Fig.  96. 

Small  size  rods  are  desirable  for  this  secondary  system,  both 
on  account  of  the  adhesion  area  and  because  of  the  ease  in  hand- 
ling the  long  lengths.  A  high  elastic  limit  steel  should  be  spe- 
cified (see  specifications  at  end  of  book) . 

Wall  Failures. — It  was  a  famous  maxim  of  Sir  Benjamin  Baker, 
that  no  engineer  could  claim  to  be  experienced  in  the  design  and 
construction  of  retaining  walls  until  he  had  several  failures  to  his 
credit.  Such,  however,  is  not  the  viewpoint  of  the  modern  engi- 
neer. It  is  to-day  clearly  apparent  that  walls,  when  they  do 
fail,  fail  for  definite  reasons  that  can  generally  be  anticipated  and 
for  which  provision  can  be  made.  It  is  necessary,  not  only  to 
find  a  proper  foundation  for  a  wall,  but  also  to  take  extreme  pre- 
caution that  such  a  foundation  will  be  maintained  permanently 
in  its  proper  condition.  It  is  essential  to  guard  against  possible 


TEMPERATURE  AND  SHRINKAGE  STRESSES     161 

saturation  of  the  bottom  and  against  erosion  of  the  soil  beneath 
the  toe  by  streams  of  water  which,  if  long  continued,  reduce  the 
bearing  capacity  of  the  soil  and  lead  to  subsequent  failure.  A 
majority  of  partial  and  complete  wall  failures  are  clearly  at- 
tributable to  foundation  weakness  developed  subsequently  to 
the  construction  of  the  wall. 

Cases  of  failure  due  to  excess  of  overturning  moment  over 
stability  moment  are  rare.  It  is  possible  that  in  placing  the  fill 
behind  the  wall,  material  may  be  dropped  from  some  height, 
either  striking  the  wall  or  setting  up  vibrations  in  the  retained 
mass  that  may  exert  an  excessive  action  upon  the  wall.  A  failure 
of  a  barge  canal  wall  in  New  York  State1  is  alleged  to  be  due  to 
this  cause.  The  fill  behind  the  wall  was  saturated  and  in  a  quak- 
ing condition.  The  material  was  dropped  behind  the  wall  by  a 
clam  shell,  from  considerable  height,  setting  up  heavy  vibrations 
in  the  mushy  mass,  which  eventually  destroyed  the  wall. 

Care  should  be  observed  in  dropping  big  stone  from  trestles  or 
from  the  partially  built  embankment  against  the  back  of  the  wall. 
While  complete  failure  is  unlikely,  small  cracks,  due  to  the  im- 
pact may  be  developed.  At  first  not  serious,  later,  due  to  frost 
and  other  weathering  action,  they 
become  unsightly,  marring  the 
face  and  eventually  develop  erosive 
gullies. 

The  improper  and  insufficient 
attention  to  drainage  (discussed  in 
a  later  chapter)  may  permit  the 
accumulation  of  water  behind  a 
wall  increasing  the  pressure  to 
such  a  degree  as  to  push  the  wall  FIG.  97. 

out  of  line. 

Among  minor  instances  of  possible  causes  of  failure,  complete 
or  partial,  may  be  mentioned  the  following. 

Lack  of  expansion  joints,  or  joints  spaced  too  far  apart. 

The  junction  of  radically  different  types  of  walls  without  a 
proper  joint.  Thus  a  wing  wall  to  an  abutment;  a  very  light 
section  wall  to  a  heavy  section  wall.  Walls  on  different  founda- 
tions. Walls  carrying  a  building  load.  A  sharp  angle  in  a 
gravity  wall,  so  that  there  is  a  component  of  the  earth  pressure 
acting  in  tension  (see  Fig.  97,  and  Photograph  Plate  No.  3a). 

1  Engineering  News,  Vol.  67,  p.  384. 
11 


162 


RETAINING  WALLS 


In  the  Trans.  Engineer's  Society  of  Western  Pennsylvania, 
Vol.  26,  it  was  noted  in  gravity  walls,  where  the  base  varied  from 
M  to  %  the  height,  that : 

"Such  failures  as  have  occurred  have  been  due,  to  the  most  part 
to  poor  construction  and  lack  of_drainage." 

In  discussing  the  action  of  clay,  both  as  a  fill  and  as  a  foun- 
dation material,  Bell,  Minutes  of  the  Proceedings,  Inst.  C.  E., 
Vol.  cxcix,  1914-5,  Part  1,  p.  233,  notes  that: 

"It  was  disquieting  to  note  the  high  percentage  of  failures  in  works 
constructed  in  clay.  Taking  all  the  available  records  of  works  subject 
to  earth  pressure,  which  had  failed,  it  appears  that  70  to  80  per  cent, 
referred  to  works  constructed  in  clay.  While  every  one  recognizes 
that  clay  is  a  treacherous  material  and  that  it  will  always  claim  a 
substantial  percentage  of  total  failures,  still  this  preponderance  is 
remarkable  and  would  perhaps  of  itself  indicate  that  there  is  something 
wrong  with  existing  methods." 

Some  Wall  Failures. —  Chas.  Baillarge1  has  pointed  out  that 
the  life  of  the  retaining  walls  in  Quebec  has  been  but  a  brief  one. 
They  were  designed  upon  the  assumption  of  a  dry  granular  fill 
and  the  base,  accordingly  was  made  from  one-fifth  to  one-third 
the  height.  Subsequently  the  filling  became  waterlogged  and 
since  no  weep  holes  or  other  drainage  had  been  provided  to  dis- 
pose of  such  accumulations  of  water,  the  excessive  pressures 
developed  caused  the  failure  of  the  walls. 


FIG.  99. 


FIG.  100. 


Mr.  Lindsay  Duncan2  has  described  the  tilting  and  settling  of 
an  abutment  prior  to  the  setting  of  the  span  upon  it.  The  sec- 
tion of  the  abutment  is  shown  in  Fig.  98.  The  wall  rested  upon 
an  adobe  foundation  and  surface  waters  gradually  softened  the 


1  Engineering  News,  Vol.  45,  p.  96. 

2  Engineering  News,  Vol.  55,  p.  386. 


TEMPERATURE  AND  SHRINKAGE  STRESSES     163 

adobe,  causing  the  wall  to  tip  forward.  An  ingenious  method 
of  reinforcing  the  wall  and  bringing  it  back  to  line  is  described 
in  the  above  article. 

Due  to  the  failure  of  a  dam1  the  foundation  of  a  wall  shown  in 
Fig.  99  was  washed  out,  and  a  section  of  the  wall  between  two 
expansion  joints  was  moved  out. 

A  wall  of  section  shown  in  Fig.  100  was  placed  in  an  old  creek 
bed.2  The  freshet  from  a  spring  thaw  undermined  the  foundation 
washing  away  the  soil  adjacent  to  the  piles.  Excessive,  loads 
developed  on  the  piles,  and  these  failed  causing  the  wall  to  settle 
about  two  feet. 

A  wall  failure  due  to  excessive  overturning  moment  is  de- 
scribed in  the  Engineering  Record,  Vol.  41,  p.  586  (see  Fig.  102). 
A  wall  of  rectangular  shape,  of  small  stone  rubble,  supported  a 


FIG.  101.  FIG.  102. 

fill  slightly  surcharged.  It  had  already  given  evidence  of  incipi- 
ent failure  by  bulging  in  several  places.  In  grading  an  adjacent 
lot,  an  additional  fill  supported  by  the  wall  "  A"  was  placed  upon 
the  old  embankment,  followed  by  the  complete  failure  of  the 
wall. 

A  wall  shown  in  Fig.  101,  supported  a  reservoir  embankment 
adjacent  to  a  roadway.3  The  brick  pavement  lining  the  road 
was  taken  up,  and  the  wall  slid  forward  from  one  to  two  feet,  and 
in  several  places  tilted  out  of  line  about  6  in.  This  seems  to 
be  an  instance  of  insufficient  frictional  resistance  between  the 
footing  and  the  wall — the  brick  pavement  supplying  the  neces- 
sary resistance  to  prevent  the  forward  movement  of  the  wall. 

1  Engineering  News,  Vol.  63,  p.  285. 

2  Engineering  News,  Vol.  61,  p.  503. 

3  Engineering  Record,  Vol.  44,  p.  7. 


PART  II 
CONSTRUCTION 

CHAPTER  VI 
PLANT 

Plant  Expenditure. — With  the  exception  of  very  small  con- 
struction jobs  amounting  to  but  a  few  hundred  dollars  in  value, 
it  is  necessary  to  employ  tools,  machinery  and  other  implements 
to  supplement  and  replace  manual  labor.  Such  auxiliary  ap- 
pliances are  termed  plant. 

There  are  no  fixed  relations  between  the  amounts  to  be  ex- 
pended on  plant  and  the  total  value  of  the  work  contemplated. 
The  principal  factors  of  a  general  nature  determining  the  amount 
of  plant  required  are,  the  yardage  of  concrete  wall,  the  time  given 
in  which  to  build  the  wall  and  the  manner  of  the  distribution  of 
the  wall  over  the  work.  Few  jobs  are  exactly  alike  or  sufficiently 
similar  that  the  plant  requirements  become  identical  and  it  is  a 
matter  of  economy  to  so  buy  plant  that  its  cost  less  its  salvage 
value,  if  any,  at  the  completion  of  the  job,  is  carried  by  this  job 
alone.  This  permits  a  careful  study  of  the  field  conditions  and 
insures  a  selection  of  plant  most  fitted  for  this  work.  It  is  a 
slogan  of  most  contractors,  that  if  a  job  is  not  worth  the  plant, 
the  job  is  not  worth  having. 

"Inasmuch1  as  plant  is  in  reality  but  a  substitute  for  labor,  it  would 
seem  obvious  that  no  more  should  be  invested  in  plant  than  will  yield 
a  good  return.  This  relation  between  plant  and  labor  is  apparently 
ignored  in  many  instances,  and  plant  charges  are  incurred  out  of  all 
proportion  to  the  volume  of  work  to  be  done.  The  ultimate  comparison, 
whether  made  directly  or  indirectly,  between  hand  labor  and  the  pro- 
posed plant,  or  between  this  and  that  plant,  must  be  made  if  the  selec- 
tion is  to  stand  the  test  of  experience. 

"The  selection  of  plant,  the  purchase  of  this  or  that  machinery,  has 
to  a  large  extent  been  more  or  less  haphazard.  Contractors  and  engi- 

1  From  "Concrete  Plant"  issued  by  Ransome  Concrete  Machinery. 

165 


166  RETAINING  WALLS 

neers,  experienced  and  successful  men,  have  been  slow  to  awake  to 
the  possibilities  for  loss  or  gain  afforded  by  plant  selection;  but  it  is 
nevertheless  deserving  of  careful  study. 

"There  seems  to  be  a  strong  tendency  toward  excess  in  plant  expendi- 
ture and  a  fact  worthy  of  note  is  the  tendency  toward  simplicity  in  plant 
upon  the  part  of  engineers  and  contractors  whose  experience  and  success 
in  the  field  entitles  them  to  be  considered  as  leaders. 

"In  estimating  plant  cost,  various  elements  other  than  first  cost  of 
plant  must  be  carefully  considered.  Cost  of  installation,  including 
freight,  cartage,  labor,  etc.,  cost  of  maintenance,  cost  of  removal,  interest 
upon  the  investment,  must  be  considered  on  the  one  hand,  as  against 
the  resultant  saving  in  labor  and  salvage  value  of  the  plant  on  the  other. 

"In  general  the  plant  best  suited  to  the  work  is  cheapest,  regardless 
of  whether  or  not  it  costs  a  few  dollars  more  than  something  less  suited 
to  the  conditions.  First  cost  is  perhaps  less  important  in  influence  on 
final  results  than  cost  of  operation  and  maintenance.  In  many  cases  a 
higher  salvage  return  will  offset  to  a  large  degree  higher  first  cost.  First 
cost,  too,  is  a  definite  constant.  It  can  be  positively  assessed  and  proper 
allowance  made  for  it  in  estimating,  in  this  respect  differing  from  main- 
tenance, which  is  an  unknown  quantity  subject  to  great  variations." 

Standard  Layouts. —  There  are  certain  types  of  work,  again, 
generally  speaking,  for  which  the  plant  layouts  are  obvious. 
Thus  a  concrete  wall  in  a  compact  area,  all  within  strategetic 
reach  of  a  center  not  exceeding  some  maximum  distance  away, 
calls  for  a  central  mixing  plant  and  a  tower  system  of  distribu- 
tion. In  track  elevation  work,  to  eliminate  grade  crossings,  the 
availability  of  a  track  adjacent  to  the  proposed  wall,  permits  the 
use  of  a  compact  concreting  train.  Usually  conditions  are  not 
so  typical  and  local  topographical  conditions,  as  well  as  the 
character  of  the  work  play  an  important  role  in  determining  the 
character  of  the  plant  best  suited  for  the  job. 

Arrangement  of  Plant. — It  may  be  stated  as  almost  axiomatic, 
that,  that  wall  is  most  economically  built  which,  other  things 
being  equal,  is  most  expeditiously  built.  This  necessitates  a 
certain  degree  of  flexibility  in  the  plant  that  little  time  may  be 
lost  in  bringing  concrete  to  the  forms  awaiting  it. 

"The  character1  and  arrangement  of  plant  depend  to  a  large  extent 
upon  local  conditions,  such  as  contour  of  ground.  The  general  layout 
of  the  work,  while  the  manner  in  which  the  materials  are  to  be  delivered 
to  the  site,  whether  in  cars  or  in  wagons,  regularly  or  irregularly,  has 
an  important  bearing  upon  the  type  of  plant.  Similarly,  the  matter  of 


PLANT 


167 


total  yardage  to  be  placed,  of  time  limit  set  for  the  work,  of  bonus  or 
penalty,  will  have  a  bearing  upon  plant  selection. 

"Other  considerations  which  may  affect  materially  the  selection  is 
the  amount  of  ground  available  for  material  storage,  and  the  time  of  the 
year  during  which  the  operation  must  be  carried  on,  winter  work  re- 
quiring very  different  plant  arrangement  from  summer  work. 

"Contour  of  ground  is  principally  effective  in  determining  the  loca- 
tion of  the  plant  with  respect  to  the  work  and  the  storage  of  materials. 
For  example,  a  steep  slope  will  often  make  advisable  a  system  of  over- 
head bins  with  gravity  feed,  which  under  other  conditions  would  not  be 
advisable. 

"The  general  layout  of  the  work  will  usually  be  the  determining  factor 
in  the  adoption  of  means  for  handling  mixed  concrete,  subject,  of  course, 
to  modifications  imposed  by  total  yardage,  etc.  It  may  make  for  the 
adoption  of  two  or  more  separate  installations  rather  than  one  central 
plant  or  it  may  cause  the  adoption  of  a  portable  plant  rather  than  a 
stationary  one. 

"Delivery  of  materials  is  principally  effective  in  determining  the 
arrangement  for  the  storage  of  raw  materials. 

"Total  yardage,  time  limit,  etc.  are  generally  the  controlling  factors 
in  determining  the  amount  available  for  plantage." 

Subdivision  of  Work.— -It  seems  natural  to  divide  the  plant 
necessary  for  concrete  retaining  walls  into  three  subdivisions: 
(1)  the  plant  to  bring  the  ma- 
terials to  the  mixer;  (2)  the 
mixer,  (3)  the  plant  to  bring  the 
materials  from  the  mixer  and 
place  it  in  the  forms. 

1.  When  the  layout  of  the 
work  is  such  that  one  or  a  few 
central  plants  may  be  used,  this 
problem  is  comparatively  sim- 
ple. The  material  is  dumped 
alongside  a  storage  bin  and  is 
fed  to  this  bin  as  required,  the  bin  having  a  hopper  to  drop 
material  into  the  mixer.  See  Fig.  103.  It  may  be  possible, 
due  to  the  advantageous  location  of  this  bin  below  the  delivery 
point,  that  the  material  cars  or  wagons  may  unload  directly 
into  the  bin.  This  requires  a  regular  and  reliable  delivery 
system  to  keep  the  bin  constantly  supplied,  since,  with  sporadic 
delivery  of  material  the  concrete  work  would  frequently  be 
delayed.  Usually  the  material  is  allowed  to  accumulate  in  a 


Storage 
Pile 


Mixer 


FIG.  103. — Loading  bin  by  derrick 
from  storage  pile  of  aggregate. 


168  RETAINING  WALLS 

storage  pile  near  the  bin  and  is  fed  from  this  pile  to  the  hopper 
bin  by  a  derrick,  with  preferably  a  clam  shell,  to  save  the  labor 
of  loading  the  skips. 

When  a  central  plant  is  not  used,  the  material  is  distributed 
along  the  site  of  the  work  in  small  piles.  It  must  be  remembered 
that  when  the  material  is  distributed  in  this  fashion,  there  is 
considerable  loss  due  to  rehandling,  to  the  gathering  of  foreign 
matter  such  as  dirt,  etc.,  and  to  the  inevitable  loss  of  the  bottom 
portion  of  the  pile  on  the  ground.  If  the  material  is  to  be  on  the 
ground  for  some  time  then  a  large  portion  of  it  may  be  lost  on 
account  of  the  weather.  Such  losses  may  amount  to  quite  a 
large  percentage  of  the  material  ordered  and  proper  allowance 
must  be  made  to  determine  the  final  net  cost  of  the  material 
in  the  concrete. 

For  this  latter  mode  of  the  distribution  of  material  the  mixer 
is  usually  fed  by  wheelbarrow  from  the  nearest  pile.  Other 
modes  of  getting  the  material  to  the  mixer  are  easily  determinable 
from  the  local  environment. 

Mixers. — The  selection  of  a  proper  mixer  is  comparatively 
simple.  The  requirements  of  good  concreting  (as  described  in  a 
later  chapter)  should  be  noted  and  a  type  of  mixer  chosen  that 
will  make  it  possible  to  carry  out  these  requirements.  The 
necessary  capacity  of  the  mixer  is  readily  determined  from  the 
expected  daily  output  required  to  prosecute  the  work  within  the 
assigned  time  limit.  Naturally  a  mixer  attached  to  a  central 
mixing  plant  if  run  continuously  will  have  a  greater  output 
than  one  of  like  capacity  carried  about  the  work.  The  catalogues 
of  the  manufacturers  of  the  various  types  of  mixers  can  be  con- 
sulted to  good  advantage  and,  with  the  advice  of  their  experienced 
salesmen,  a  type  most  suitable  for  the  work  can  readily  be  selected. 

"It1  is  true  that  one  mixer  may  have  an  excess  of  power  with  resultant 
acceleration  of  the  various  operations  going  to  complete  the  mixing 
cycle,  one  machine  may  be  quicker  in  mixing  or  discharging  than  another; 
but  these  differences  will  influence  the  final  result  less  than  a  defective 
organization.  For  example,  it  is  common  practice  to  employ  extra  men 
to  fill  wheelbarrows,  a  practice  which  increases  the  cost  of  this  work 
twenty-five  to  thirty-five  per  cent,  according  to  whether  or  not  the 
wheeler  helps  fill  his  own  barrow.  Similarly  it  is  common  practice  to 
handle  mixed  concrete  in  small  wooden  or  iron  barrows  holding  an  average 
of  two  cubic  feet.  By  furnishing  substantial  runways  and  the  adoption 

*  Ibid. 


PLANT  169 

of  carts  an  average  load  of  4.5  cubic  feet  can  easily  be  handled.  It  is 
to  such  elements  of  organization  that  attention  should  be  directed,  if 
you  would  cut  down  the  cost  of  operation.  Properly  handled,  concrete 
plant  becomes  an  important  factor  in  setting  the  pace  for  the  work. 

"Cost  of  installation  includes  freight,  cartage  and  erection,  elements 
varying  with  the  character  of  the  plant,  location  of  the  work,  with 
respect  to  the  source  of  supply,  etc.  *  *  *. 

*  *  *  ti  NO  other  class  of  machinery  is  subjected  to  the  severe  usage 
imposed  on  concrete  machinery.  The  nature  of  the  materials  handled 
make  for  excessive  wear,  to  which  should  be  added  the  fact  that  the 
machinery  is  ordinarily  handled  by  a  class  of  labor  not  calculated  to 
give  it  the  intelligent  care  and  attention  to  which  it  is  properly  entitled. 
It  is  to  long  experience  upon  the  part  of  the  manufacturer  in  this  special 
field  that  the  purchaser  must  look  for  protection  against  failure,  under 
the  severe  conditions  which  actually  prevail  in  the  field.  The  history  of 
success  in  this  line  of  work  is  a  history  of  constant  changes  in  design, 
a  story  of  heavier,  stronger  parts,  of  adapting  the  machine  to  the  character 
of  the  work  by  reducing  parts  to  a  minimum. 

"The  fewer  parts  your  machine  has,  the  less  likely  it  is  to  get  out  of 
order,  and  the  more  readily  the  operator  of  ordinary  capacity  can  keep 
it  in  working  order. 

"Considered  broadly,  mixers  may  be  divided  into  Drum  Mixers, 
Trough  Mixers,  Gravity  Mixers,  Pneumatic  Mixers. 

"Drum  Mixers  may  again  be  divided  into  Tilting  Mixers  (Smith  Type) 
and  Non-Tilting  (Ransome  Type).  In  the  former  class  the  mixing 
drum  is  mounted  on  a  swinging  frame,  and  the  discharge  of  the  mixed 
materials  is  accomplished  by  a  tipping  of  the  frame  and  drum.  In  the 
latter  class  mixed  materials  are  drawn  out  through  a  chute  inserted  in 
the  drum. 

"Trough  mixers,  as  a  whole,  may  be  designated  as  Paddle  Mixers, 
though  the  paddles  may  vary  in  form  from  a  broken  worm,  through  the 
various  stages,  to  the  continuous  worm  and  the  conveyor  flight  may  be 
single  or  double,  of  varying  or  uniform  pitch. 

"Gravity  Mixers  are  of  the  same  general  characteristics,  depending 
for  success  upon  a  series  of  deflectors,  chains,  pegs,  or  conical  hoppers,  for 
the  mixing  action.  They  are  not  adapted  to  building  work  in  any  case 
and  do  not  deserve  serious  consideration  here. 

"Pneumatic  Mixers  include  the  various  types  of  pneumatic  mixers 
developed  during  the  past  two  or  three  years  by  Wm.  L.  Canniff,  A.  W. 
Ransome,  McMichael,  Eichelberger.  In  the  Ransome  and  Canniff 
mixers,  the  materials  are  first  mixed  by  air  in  a  container,  and  the  mixed 
concrete  then  forced  out  through  pipes  to  its  ultimate  destination. 
In  the  McMichael  and  Eichelberger  machines  the  materials  are  assem- 
bled in  a  container  and  forced  through  pipes  without  premixing.  These 
latter  machines  depend  for  successful  results  upon  such  mixing  action 


170 


RETAINING  WALLS 


as  may  take  place  in  transit  through  the  pipe.  Pneumatic  mixers  are 
all  expensive  to  operate  and  cannot  be  used  to  advantage  except  in 
special  cases." 

Distributing  Systems. — There  is  greater  latitude  in  the  selec- 
tion of  plant  for  a  distributing  system  than  in  the  selection  of 
plant  for  the  two  prior  operations  and  since  this  portion  of  the 
work  is  the  most  costly  of  the  three,  greater  care  should  be  spent 
upon  the  proper  selection  of  the  necessary  plant. 

A  retaining  wall  covers,  generally,  a  long  narrow  strip,  making 
a  compact,  single  distributing  system  from  a  single  central  plant 
usually  out  of  the  question.  Nevertheless,  heavy  walls,  with 
large  concrete  yardages  within  fairly  restricted  areas  may  permit, 
economically  the  use  of  one  or  more  central  distributing  plants. 


Tower 

•Mixer 
FlafCar 


J77777\  tyflTTi 

FIG.  104. — Pouring  concrete 
by  tower  and  mixer  mounted 
on  flat  car. 


FIG.  105. — Pouring  concrete  from 
platform  erected  on  trestle. 


The  greater  mass  of  the  wall  lying  above  the  ground  surface, 
the  concrete  must  be  raised  to  permit  its  placement  within  the 
forms.  This  is  accomplished  by  several  methods.  The  mixer,  a 
travelling  one,  may  be  raised  and  its  contents  spouted  directly,  by 
gravity,  into  the  form.  The  mixer  may  remain  on  the  ground 
and  its  contents  raised  and  delivered  into  the  form.  Following 
are  some  possible  methods  of  this  latter  mode  of  distribution. 

(a)  The  mixer  is  on  a  flat  car,  with  a  tower  and  hoist  (see 
Fig.  104). 

(6)  The  mixer  is  on  the  ground  and  the  concrete  taken  from  it 
by  cars,  or  barrows  and  run  over  platforms  along  the  top  of  the 
form  into  the  wall  (see  Fig.  105). 

(c)  A  derrick  takes  the  bucket  from  the  mixer  and  dumps  its 
contents  either  directly  into  the  form  or  into  a  spouting  device 
leading  to  the  form. 

(d)  Tower  distribution. 

(e)  Cableway  distribution. 
(/)  Pneumatic  distribution. 


PLANT  171 

"The  handling1  of  concrete  through  spouts  or  chutes  is  of  compara- 
tively recent  development,  and  as  in  many  other  similar  developments, 
there  has  been  a  tendency  to  overdo.  Spouting  systems  have  been 
installed  on  many  buildings  where  the  distribution  might  have  been 
better  done  by  barrow  or  cart. 

"The  installation  of  a  spouting  system  is  expensive,  and  should  not 
be  undertaken  blindly,  nor  with  expectations  of  abnormal  savings. 

"Spouting  plants  may  be  grouped  under  Boom  plants,  Guy  line 
plants,  Tripod  plants.  In  the  former,  the  spouting  is  mounted  on  a 
swivelled  bracket  at  the  tower  end,  and  the  outer  end  supported  by  a 
boom  moves  freely  about  the  work.  A  second  length  of  spout  ordinarily 
completes  the  unit.  This  type  of  plant  has  a  greater  freedom  of  move- 
ment than  either  guy  line  or  tower  plants,  but  is  not  as  free  moving  as 
might  be  desired. 

"Many  means  have  been  tried  to  facilitate  ready  moving  of  the  free 
end,  none  of  them,  however,  proving  entirely  satisfactory.  A  sugges- 
tion has  been  made  to  counterbalance  the  free  end,  but  this  has  not, 
as  yet,  been  tried  out  thoroughly. 

"In  guy  line  plants,  the  spouting  is  suspended  by  ordinary  blocks 
and  falls  from  guy  lines  or  from  special  cables  set  up  for  the  purpose. 
In  some  cases  the  outer  end  of  the  cable  is  mounted  on  a  portable  tower 
or  "A"  frame  and  the  blocks  and  falls  are  preferably  arranged  so  that 
necessary  adjustments  in  the  line  may  be  made  from  the  ground. 

"In  Tripod  plants  movable  towers  are  used  to  support  the  ends  of 
various  sections  of  spouting. 

"It  has  been  found  by  practical  experience  that  concrete,  thoroughly 
mixed  and  of  proper  consistency  will  flow  on  a  slope  of  eighteen  degrees, 
with  the  best  results  obtained  at  twenty-three  degrees.  These  slopes, 
however,  are  based  upon  a  rigidly  supported  chute.  Where  the  spouts 
are  supported  from  guy  lines,  the  slope  must  be  a  little  steeper,  prefer- 
ably from  twenty-seven  to  thirty  degrees.  By  proper  consistency  is 
meant  a  mixture  with  approximately  one  and  a  quarter  to  one  and  a 
half  gallons  of  water  to  the  cubic  foot  of  material.  There  should  be 
just  as  much  water,  as  the  material  can  carry  without  separation, 
so  that  the  stone  particles  will  be  carried  in  suspension  in  the  mass. 
There  should  be  a  sliding  of  materials  down  the  spout  rather  than  a 
rolling. 

"Various  types  of  spouting  have  been  tried,  ranging  from  round  pipe 
to  rectangular  troughs.  Best  results  have  been  secured  from  the  use 
of  5-inch  pipes,  or  10-inch  open  troughs,  the  latter  having  the  preference 
for  flat  slopes  and  the  former  where  there  is  necessity  for  varying  pitch, 
some  of  steeper  pitch  than  named  above. 

"With  open  spouting,  the  use  of  line  hoppers  in  connection  with 

1  "Concrete  Plants,"  Ransome  Concrete  Machinery,  p.  23. 


172  RETAINING  WALLS 

flexible  spouting  accomplishes  satisfactorily  the  necessary  changes  in 
pitch.  The  greatest  items  of  expense  in  spouting  plants  are  first  cost, 
installation  and  maintenance. 

"Maintenance  charges  are  particularly  heavy.  The  ordinary  stock 
spouting  which  is  made  of  No.  14  gage  metal  will  seldom  handle  more 
than  two  thousand  yards  without  renewal .  This  is  due  to  the  abrasive 
action  of  the  material,  especially  as  affecting  the  rivets  which  join  the 
various  sections. 

"In  general  we  would  say  that  whether  or  not  you  can  use  spouting 
to  advantage  must  be  carefully  considered  for  each  job.  Where  the 
work  is  light  and  scattered  any  attempt  to  spout  concrete  into  place 
is  foredoomed  to  failure." 

"The  economy1  of  distributing  concrete  through  properly  designed 
chuting  plants  has  been  demonstrated  time  after  time,  on  all  kinds 
of  construction  and  it  has  been  conclusively  shown  that  properly 
proportioned,  thoroughly  mixed  concrete  may  be  conveyed  to  any 
mechanically  practical  distance  without  disintegrating  the  mass. 

"Concrete  should  flow  at  a  uniform  speed  of  from  seventy-five  to 
one-hundred  feet  per  minute.  The  best  results  are  attained  with  the 
chute  line  pitched  with  a  fall  of  one'  foot  in  four  up  to  150  feet  radius. 
For  longer  distances  the  fall  should  be  about  one  in  three,  starting 
with  one  foot  in  four  and  increasing  the  grade  towards  the  discharg- 
ing point." 

When  it  is  remembered  that  a  cableway  mode  of  distribution 
moves  in  but  two  dimensions  i.e.  in  a  vertical  plane  only  and  that 
its  cost  rapidly  increases,  and  the  amount  of  load  to  be  carried, 
decreases  with  an  increase  in  span,  its  use  as  a  distributing  system 
is  usually  discarded  for  the  methods  of  distribution  previously 
mentioned. 

Below  are  given  a  series  of  descriptions  of  various  plants 
used.  While  it  is  impractical  to  attempt  to  make  a  standard 
classification  of  construction  problems,  the  illustrations  selected 
are  thought  to  be  more  or  less  typical  and  the  character  of  the 
plant  used  probably  the  most  fitted  for  the  environment  and 
character  of  the  work  at  hand. 

(A)  TOWER  DISTRIBUTION 

Railroad  station  at  Memphis  *  *  *  111.  Cent.  R.  R.  (see  Fig. 
106)  for  the  skeleton  layout  of  the  work),  Engineering  News,  Vol. 
72,  p.  629. 

"The  construction  of  the  retaining  walls  and  subway  bridges  was 
hampered  by  the  necessity  of  providing  for  traffic.     There  were  about 
1  Bulletin  No.  23,  The  Lakewood  Engineering  Co. 


PLANT  173 

60  trains  daily,  the  heaviest  traffic  being  from  7  A.M.  to  noon  and  3  to 
5  P.M.  The  only  freight  movements  over  this  part  of  the  line  were  in 
switching  service.  The  great  difficulties  encountered  were  the  limited 
space  available,  the  handling  of  concrete  while  keeping  clear  of  the  trains 
and  the  inability  of  the  contractor  to  get  certain  parts  of  the  site  delivered 
to  him  for  work  at  the  time  desired.  For  all  work  *  *  *  the  storage 
space  for  materials  was  limited  and  it  was  necessary  to  regulate  ship- 
ments of  all  kinds  so  as  to  be  able  to  use  the  material  upon  arrival. 

(<-•  -----  2400'  - .j 


FIG.  106. — Layout  of  retaining  walls  and  abutments. 

"The  concrete  was  delivered  in  place  by  spouting  from  elevator 
towers,  using  self-supporting  trussed  chutes.  Two  stationary  plants 
with  100'  towers  and  one  portable  plant  with  a  50'  tower  were  used, 
each  of  the  former  being  set  up  twice  (in  different  locations)  and  the 
latter  being  shifted  as  required.  Each  had  its  mixer,  and,  in  order  to 
work  to  full  capacity,  a  two-compartment  material  bin  or  hopper  was 
erected  over  the  mixer,  holding  about  30  cubic  yards  of  stone  and  15 
cubic  yards  of  sand.  The  materials  were  brought  in  railway  cars 
and  unloaded  direct  to  the  mixer  bin  or  to  small  storage  piles,  there 
being  little  room  for  storage.  A  derrick  with  clam  shell  bucket  took 
the  material  from  the  car  or  storage  pile  and  dumped  it  into  1^  cubic 
yard  cars,  which  were  hauled  up  a  cable  incline  and  dumped  into  the 
material  hopper.  The  incline  had  a  four  rail  track  in  the  lower  portion 
and  a  three  rail  track  at  the  top.  *****  The  maximum  output  per 
day  was  550  cubic  yards.  The  entire  concrete  yardage  was  30,000 
cubic  yards." 

(B)  CONCRETE  TRAINS 

As  has  been  previously  noted,  railway  improvement  work,  such 
as  track  elevation  or  depression,  permits  the  use  of  a  compact 
concrete  train.  A  typical  piece  of  work  is  the  track  elevation 
work  of  the  Rock  Island  lines,  described  in  Engineering  News 
Vol.  73,  p.  670;  Vol.  74  p.  1275  and  Vol.  74  p.  890.  The  con- 
crete plant  which  placed  the  necessary  30,000  yards  of  concrete 
for  this  improvement  is  described  as  follows : 

"  Concrete  train  consists  of  a  mixer  car,  four  to  seven  stone  cars  and 
two  to  four  cars  of  sand.  ******  The  mixer  car  is  a  thirty-five 
foot  flat  car,  equipped  with  a  %  yard  Smith  non-tilting  mixer  10  h.p. 


174  RETAINING  WALLS 

vertical  engine,  20  h.p.  vertical  boiler,  700  gallon  storage  tank  and  60 
gallon  feed  tank  for  the  mixer.  The  machinery  is  housed  the  roof  of  the 
car  being  higher  at  the  discharging  hopper  than  at  the  ends  of  the  car, 
thus  forming  an  easy  incline  from  the  runways  on  the  tops  of  the  gondola 
cars  to  the  charging  hopper  above  the  mixer.  The  mixer  is  located 
about  8'  from  one  end  of  the  car  and  faces  the  end.  It  discharges 
the  concrete  into  a  swivelling  chute  which  may  be  swung  to  discharge 
the  end  or  either  side  of  the  car.  This  arrangement  of  pouring  from 
different  angles  or  from  either  end  of  the  train  eliminates  the  necessity 
of  turning  the  mixer  car  (as  required  with  the  other  types)  and 
makes  a  considerable  saving  in  working  train  space. 

•'The  chute  has  intermediate  openings,  so  that  concrete  can  be  dis- 
charged at  different  points.  A  man  on  top  of  the  car  regulates  the 
charging  of  the  mixer,  the  supply  of  water  and  the  dumping  of  the 
concrete  (see  Fig.  107).  Usually  the  mixer  train  stands  on  trestles 

Steam  Donkey 

}    Loading 

•      Chu+e-.  f  Operators  Platform 


OQ       QO 


FIG.  107. — Connecting  train. 

and  the  concrete  is  spouted  to  the  form  beneath.  For  the  upper  part 
of  the  piers,  it  has  been  necessary  to  elevate  the  concrete,  a  crane  and 
bucket  being  used  to  place  the  concrete  in  the  forms. 

"The  mixer  is  designed  to  carry  a  tower  and  hoisting  engine  if  re- 
quired. *******  ^  valuable  feature  of  the  car  is  a  powerful 
winch-head  for  a  cable,  which  is  anchored  ahead.  This  enables  the 
mixer  car  to  move  the  train  along  as  the  work  progresses,  thus 
dispensing  with  the  constant  attendance  of  locomotive  and  crew. 

"Each  train  is  placing  at  the  rate  of  20  to  30  cubic  yards  per  hour, 
with  a  monthly  total  for  both  trains  of  11,000  yards  of  concrete." 

Other  instances  of  the  use  of  similar  work  trains  are  mentioned 
below. 

Engineering  News,  Vol.  75,  p.  634.  In  filling  in  an  old  trestle,  and  build- 
ing the  necessary  retaining  walls,  a  concrete  train  of  three  cars  one  mixing, 
one  stone  and  one  sand,  were  used. 

Engineering  News,  Vol.  75,  p.  1192.  The  interesting  feature  of  the  work 
train  here  was  the  fact  that  the  hoist  was  operated  by  steam  from  the  loco- 
motive. 

Engineering  News,  Vol.  75,  p.  494.  Fort  Wagner  Track  Elevation.  The 
concrete  train  worked  on  a  temporary  operating  trestle,  the  track  being 
out  of  commission  while  the  concrete  train  was  on  it. 

Engineering  Record,  Vol.  70,  p.  240.     Chicago,  Milwaukee  and  St.  Paul. 


PLANT  175 

The  concrete  train  operated  upon  a  trestle.  A  cableway  on  the  concrete 
train  took  materials  from  the  intermediate  cars  to  the  bins.  This  proved 
cheaper  than  tower  cars  and  hoist  cranes. 

Cableway. — The  use  of  a  cableway  for  pouring  the  concrete  walls 
of  a  viaduct  is  described  as  follows  in  the  Engineering  News, 
Vol.  72,  p.  930  (see  Fig.  108). 

''Concrete  material  was  delivered  in  cars  on  a  siding  and  unloaded 
unto  stock  piles  by  a  stiff-leg  derrick  mounted  (with  its  engine  and 
hoist)  on  a  tower  or  platform  some  15'  high.  The  same  derrick  and 
clam  shell  bucket  handled  the  material  from  the  stock  piles  to  the  200 
yard  bins  over  the  one-yard  concrete  mixing  plant  which  was  located 
just  east  of  the  structure  and  on  the  north  side  of  the  tracks. 


*" 

-Railroad  Tracks 

H 

\ 

<  Mixing  ra 
Plant-* 
U-  —  

\ 

...  HCttJ—- 

FIG.  108. — Layout  for  cableway. 

"The  cableway  was  800  feet  long  with  an  80  foot  tower  at  the  mixer 
end  and  a  single  bent  60  feet  high  at  the  further  end.  It  was  placed  over 
each  wall  in  turn  and  was  shifted  laterally  80  feet,  from  one  wall  to 
the  other  without  being  dismantled;  this  was  done  by  placing  timber 
dollies  under  the  tower.  Handling  the  12,500  yards  of  concrete  by 
cableway  was  economical  as  the  amount  of  concrete  at  the  ends  of  the 
walls  is  small  and  wheeling  it  in  buckets  would  have  been  slow  and 
expensive." 

An  interesting  comparative  analysis  of  the  use  of  several 
different  plant  layouts  for  a  series  of  similar  pieces  of  work  is 
described  by  Mr.  Armstrong  in  the  Journal  of  the  Western  Soc.  of 
Engineers,  Vol.  16.  New  Passenger  Terminal:  C.  &  N.  W.  R.  R. 

The  retaining  walls  enclosed  a  rectangular  layout,  bounded 
by  two  street  crossings  and  the  parallel  easement  lines. 

The  plant  layouts  to  pour  the  walls  were  as  follows : 

(a)  A  cableway,  placed  on  movable  trucks  was  used,  permit- 
ting the  shifting  of  the  towers  to  pour  each  of  the  walls.  This 
plant  did  not  prove  economical  and  was  of  low  capacity.  The 
best  run  was  24  yards  per  hour. 

(6)  A  runway  with  rails  ran  around  the  top  of  the  wall  forms. 
A  derrick  hoisted  the  buckets  of  concrete  to  a  hopper  which 


176  RETAINING  WALLS 

dumped  into  cars  running  along  the  form  runway.  This  was 
cheaper  than  the  cableway  and  had  a  capacity  of  about  33  yards 
per  hour. 

(c)  In  place  of  the  derrick  as  above  a  short  tower  was  used 
with  a  hoisting  engine.     The  best  average  was  37  yards  per  hour. 
The  dump  cars  ran  as  much  as  500  feet  away  from  the  tower. 

(d)  A  mixer,  elevator  and  a  hoist  were  mounted  on  a  car  and 
ran  around  the  forms.     This  proved  very  unwieldy  and  could 
not  get  close  to  the  forms.     Less  labor  was  needed  here,  however, 
since  the  dump  cars  were  eliminated.     The  best  results  with  this 
plant  were  about  25  yard  of  concrete  per  hour. 

The  following  is  a  trite  recommendation  by  the  author  of  the 
above  paper: 

"It  might  be  stated  as  a  general  principle  in  the  design  of  plant 
that  the  capacity  of  the  mixer  should  be  made  the  determining  factor 
in  the  output.  The  charging  hoisting  and  conveying  appliances  should 
be  designed  with  such  a  degree  of  flexibility  as  to  prelude  the  possibility 
of  retarding  the  mixing  process  by  delay  in  charging  the  mixer  or  delay 
in  removing  the  discharged  concrete.  The  mos-t  economical  mixer, 
other  things  being  equal,  is  the  one  which  discharges  its  mixed  batch 
and  receives  its  new  batch  in  the  shortest  time." 

Tower  and  Trestle.1 — In  concreting  a  high  wall,  50  feet  in 
height,  the  following  description  is  given  of  the  plant  used. 

Storage  Bin 
and  Mxer 


Railroad 


Trestle  along 
••'  Wall 


Fio.   109. — Central  mixing  plant.     Combined  tower  and  trestle  distribution. 

"For  concreting  the  wall  a  very  efficient  plant  was  installed.  A  Hains 
gravity  mixer  was  located  about  the  center  of  the  length  of  the  wall, 
where  it  was  easily  loaded  by  derrick,  from  the  adjacent  high  level 
railway.  Concrete  from  the  bottom  or  delivery  end  of  this  mixer 
was  run  into  an  elevator  whence  it  was  lifted  to  be  dumped  into  a 
hopper  and  chute  leading  to  another  hopper  with  a  bottom  dump  located 
on  a  frame  just  outside  of  the  wall  forms.  All  of  the  preceding  equip- 

1  Engineering  News,  Vol.  73,  p.  776. 


PLANT 


111 


ment  was  stationary,  but  alongside  of  the  wall  was  a  trestle  which  took 
concrete  from  the  last  noted  hopper  and  dumped  it  through  another 
chute  to  its  proper  place  in  the  forms  (see  Fig.  109).  The  number 
of  chutings  given  each  batch  should  be  especially  noted." 

In  pouring  a  retaining  wall  for  the  Baltimore  and  Ohio  Im- 
provements1 the  inaccessibility  of  the  site  made  it  necessary  to 
use  a  gantry  crane  device  with  a  platform  and  stiff  leg  derrick, 
as  shown  in  Fig.  110.  A  narrow  gage  railroad  ran  alongside 
the  roadway  and  brought  the  concrete  from  a  central  mixing 
plant  about  one-half  a  mile  from  the  work.  The  gantry  served 
also  to  support  the  wall  forms.  (This  work  is  also  described  on 
page  211  under  winter  concreting.) 


Steel 
Forms 


«     rH 

T/775 
^7777777777) 


J I LJL 


Railroad  in 
Operation 


Construction 
Dinkey  Line 


FIG.  110. 

The  following  is  an  interesting  description  of  several  methods 
of  handling  the  material  on  a  bridge  abutment  job.2 

"Hopper  cars,  derrick  skips,  elevator  buckets  and  inclined  chutes 
were  combined  in  placing  3360  cu.  yds.  of  concrete  in  abutments  and 
approach  retaining  walls  for  a  steel  highway  bridge  across  the  Chicago 
&  Northwestern  Ry.  at  Wheaton,  111.  To  give  increased  headway  the 
bridge  is  at  a  higher  elevation  than  the  old  span  parallel  to  it,  so  that 
long  inclined  approaches  were  required,  practically  at  right  angles 
to  the  bridge,  as  shown  by  the  accompanying  plan  (see  Fig.  111). 
Each  approach  has  a  retaining  wall  on  one  side,  and  the  wall  on  the 
south  side  of  the  railway  is  about  600  feet  along. 

"A  concrete-mixing  plant  was  located  beyond  the  end  of  the  cut. 
Sand  and  gravel  were  unloaded  from  cars  into  stock  piles  on  the  side 
of  the  adjacent  fill,  and  the  stone  was  loaded  into  an  elevated  bin 
by  a  derrick  with  a  grab  bucket.  The  sand  was  wheeled  to  the  loading 
chute.  The  mixer  discharges  the  concrete  into  a  sidegate  hopper  car. 


1  Engineering  News,  Vol.  76,  p.  269. 

2  Engineering  News-Record,  March  13,  1919,  p.  553. 

12 


178 


RETAINING  WALLS 


PLANT  179 

"  Between  this  plant  and  the  bridge  site  an  elevator  tower  with  a 
chute  was  erected,  while  beyond  this  and  close  to  the  abutment  was  a 
guyed  derrick,  both  tower  and  derrick  being  on  the  narrow  strip  between 
the  old  road  and  the  top  of  the  cut.  A  narrow-gage  track  with  one 
automatic  siding  extended  from  the  mixer  plant  to  the  tower  and  derrick. 
This  was  operated  by  an  endless  cable  with  a  hoisting  engine  placed  near 
the  derrick  and  on  it  the  concrete  was  handled  in  the  hopper  cars  men- 
tioned above. 

"At  first  the  concrete  was  delivered  to  the  elevator  buckets  and 
spouted  to  the  forms.  The  tower  chute  or  spout  extended  across  the 
road  and  delivered  the  concrete  into  lateral  chute  supported  directly 
above  the  forms  by  falsework.  This  sufficed  for  about  one-half  the 
length  of  the  wall. 

"For  the  remainder  of  the  work  the  cars  ran  up  to  the  derrick  and 
discharged  the  concrete  into  a  home-made  wooden  skip  which  was  placed 
in  a  pit  at  the  side  of  the  cable  track  and  was  handled  by  the  derrick. 
A  movable  gate  was  fitted  to  one  end  of  the  skip,  with  inclined  boards 
on  the  inside  to  guide  the  concrete  to  the  opening  and  to  prevent  it 
from  being  pocketed  in  the  corners.  The  skip  was  dumped  into  a  feed 
hopper  at  the  summit  of  the  inclined  chutes  carried  along  and  above  the 
forms  for  falsework. 

"Concrete  for  the  abutment  on  this  side  of  the  railway  was  placed 
directly  by  the  derrick  and  skip.  For  the  abutment  and  short  wall  on 
the  opposite  side  and  inclined  chute  was  extended  across  the  tracks, 
having  a  feed  hopper  at  its  upper  end  within  reach  of  the  derrick.  At 
its  lower  end  was  a  vertical  drop  line  leading  to  the  head  of  the  chutes 
over  the  abutment  form,  these  being  shifted  to  deliver  the  concrete 
in  the  desired  portions  of  the  form. 

"Baffles  were  used  at  the  discharge  ends  of  the  long  chutes  to  prevent 
segregation  of  the  concrete  as  it  was  deposited  in  place.  In  some  cases 
these  were  short  troughs  secured  to  the  trench  bracing  or  form  struts, 
being  placed  opposite  the  end  of  the  chute  and  sloping  in  the  opposite 
direction,  so  that  the  direction  of  the  concrete  was  reversed  just  before 
its  final  discharge." 

Conclusion. — To  summarize,  plant  is  employed  solely  to  effect 
an  economy  in  the  construction  of  a  wall.  To  use  plant  that 
does  not,  in  the  final  analysis,  show  a  saving  because  of  its  em- 
ployment, is  unjustifiable.  It  is  understood,  of  course,  that  all 
economies  accomplished  are  legitimate  ones;  not  such  as  are 
made  at  the  expense  of  good  construction. 

Bearing  in  mind  that  most  jobs  are  unique  in  character,  plant 
should  be  bought  for  the  sole  requirements  of  the  work  at  hand 
and  in  proportion  to  the  total  cost  of  the  work.  Such  illustra- 


180  RETAINING  WALLS 

tions  of  actual  construction  work  as  have  been  cited  may  furnish 
an  idea  of  general  plant  layouts — but  each  piece  of  work  contem- 
plated must  be  studied  out  individually  that  advantage  may  be 
taken  of  all  local  situations,  such  as  topography,  railroad  and 
highway  location  and  the  like. 

Naturally  some  pieces  of  plant  are  standard.  A  good  mixer, 
hoists,  derricks  and  small  plant  such  as  barrows,  carts,  shovels, 
etc.,  may  survive  a  job  and  be  easily  fitted  to  other  work.  This 
is  a  matter  of  judgment.  Little  mistake  is  made,  however,  if 
plant  is  procured  for  one  job  and  charged  off  to  that  one  job. 
The  cost  accounting  and  the  preparation  of  bids  for  new  work 
are  thus  vastly  simplified  and  each  job  carries  itself,  the  ideal 
contracting  condition. 

In  the  following  chapters  some  stress  is  laid  upon  the  require- 
ments of  good  form  work  and  of  good  concrete  work.  To  secure 
the  proper  results  as  indicated  in  those  chapters  requires  a  co- 
ordination between  the  plant  and  the  methods  used  and  plant 
that  will  make  it  difficult  to  secure  the  desired  results  should  not 
be  employed.  It  is  only  just  to  add  that  plant  manufacturers 
are  keenly  alive  to  the  demands  of  modern  construction  and  strive 
to  cooperate  with  the  engineer  and  contractor  to  supply  ma- 
chinery that  will  aid  in  turning  out  flawless  work. 

Plant  Literature 

Ransome  Concrete  Machinery  Co.,  "Concrete  Plant." 

HOOL,  "Reinforced  Concrete,"  Vol.  II. 

TAYLOR  and  THOMPSON,  "Concrete  Costs,"  pp.  376-380. 

"Handbook  of  Construction  Plant,"  R.  T.  DANA. 

"Concrete  Engineers  Handbook,"  HOOL  and  JOHNSON,  "Concreting  Plant." 


CHAPTER  VII 
FORMS 

Panels. — Form  work  for  concrete  walls  may  be  divided  into 
two  parts,  (a)  the  form  panel  proper,  consisting  of  the  lagging 
with  the  supporting  joists  and  (6)  the  necessary  bracing  to  hold 
the  form  panel  in  place.  With  the  exception  of  very  small  jobs 
or  of  intricate  and  varying  shaped  walls,  forms  are  usually  de- 
signed to  be  used  several  times.  To  insure  maximum  economy, 
then,  it  is  necessary  that  the  panels  be  stoutly  built,  yet  of  such 
dimensions  that  they  be  easily  set  up,  stripped  and  carried  about. 
The  details  should  be  such  that  the  panels  can  be  assembled,  put 
in  place  and  made  grout  tight  with  a  minimum  of  carpentry  work. 

Concrete  Pressure. — That  the  form  panel  be  properly  de- 
signed, it  is  necessary  that  some  attempt  be  made  to  determine 
the  amount  of  the  concrete  pressure.  Both  theoretically  and 
experimentally,  it  has  been  found  exceedingly  difficult  to  formu- 
late the  action  of  wet  concrete  upon  the  form.  At  the  instant  it 
is  placed  in  the  form,  its  pressure  approximates  closely  a  fluid 
pressure,  the  fluid  weighing  150  pounds  per  cubic  foot.  Soon 
afterwards,  both  on  account  of  the  setting  action  and  of  the  solids 
contained  in  the  concrete,  the  pressure  drops  away  from  the 
linear  fluid  pressure  law.  .  For  a  thin  wall  with  the  concrete 
level  rising  with  a  fair  degree  of  rapidity,  this  linear  law  (  p  =  wh) 
is  a  good  approximation.  For  a  wall  of  heavy  section,  such  as  a 
gravity  wall  and  the  like,  this  linear  law  would  give  excessive 
pressures. 

Concrete  pressures  are  quite  often  underestimated  with  the 
result  that  the  forms  yield,  or  give  way  entirely,  spoiling  much 
work  and  entailing  an  expense  far  in  excess  of  that  required  by 
the  increased  amount  of  material  to  hold  the  concrete  properly. 

Probably  the  most  extensive  series  of  experiments  upon  con- 
crete pressures  and  the  one  most  frequently  quoted,  were  those 
performed  by  Major  Shunk.1  His  conclusions  are  as  follows: 

1  A  re'sume'  of  these  experiments  is  given  in  Engineering  News,  Vol.  62,  p. 
288. 

181 


Temp. 


182  RETAINING  WALLS 

The  pressure  of  concrete  follows  the  linear  law 

p  =  wh  (182) 

with  w  equal  to  150  Ib.  per  cubic  foot,  until  a  time  T  has  elapsed, 
in  minutes, 

T  =  c  +  150/E  (183) 

where  c  is  a  constant  depending  upon  the  temperature  of  the 

mix  (see  Table  31)  and  R  is  the  rate  of 
TABLE  31. — CONCRETE  .  ,,  1-1,1 

PRESSUKE  CONSTANTS        POUrmS  l'e'  the  rate  at  whlch  the  con- 
crete  is  rising  in  the  form,  in  feet  per 

c  hour.      A    series   of   charts  giving  the 

pressure  after  the  time  T  has  elapsed  is 
given  in  the  re"sum£  of  the  report  quoted 

i  U  ^o  i 

60  35  above' 

55  42  A.    series    of   experiments   upon   the 

50  50  pressure   of  liquid   concrete    has    been 

40  70  given  by  Hector  St.  George  Robinson. 

See  Minutes  of  the  Proceeding  of  the 

Institute  of  Civil  Engineers,  Vol.  clxxxvii,  1911-1912,  Part  1, 
"The  Lateral  Pressure  of  Liquid  Concrete"  excerpts  of  which 
are  quoted  here: 

"Numerous  experiments  were  made  on  different  types  of  concrete 
structures.  In  heavy  walls,  large  piers  and  other  members  of  fair  size 
the  lateral  pressure  exerted  was  found  to  be  fairly  uniform  and  practically 
constant  for  equal  heads;  but  in  reinforced  concrete  columns  of  small 
dimensions,  thin  walls  and  other  light  concrete  work,  the  effect  of  fric- 
tion between  the  more  or  less  rough  timber  forms  and  the  concrete,  to- 
gether with  the  arching  action,  was  found  to  reduce  the  pressure 
considerably. 

"The  first  series  of  tests  were  made  during  the  building  of  a  long  wall 
about  three  feet  thick,  constructed  of  concrete  weighing  140  pounds 
per  cubic  foot  and  composed  of  slow-setting  cement,  sand  and  crushed 
granite  in  the  proportions  of  1  : 3  : 6  by  volume.  In  mixing  sufficient 
water  was  used  to  bring  it  to  a  thoroughly  plastic  condition,  requiring 
little  or  no  tamping  to  consolidate.  The  concrete  was  laid  more  rapidly 
than  is  usual  in  this  class  of  work,  being  carried  up  as  rapidly  as  the 
mixing  and  placing  would  permit  to  a  height  of  8  feet  above  the  center 
of  the  pressure  face,  during  which  time  a  light  iron  bar  with  a  turned 
up  end  was  used  for  churning  the  semi-liquid  mass. 

"  The  second  series  was  carried  out  on  large  piers,  four  feet  square,  the 
concrete  in  this  case  being  a  1  : 2  : 4  mixture  of  cement,  sand  and  Thames 
ballast,  weighing  about  145  Ibs.  per  cubic  foot.  The  conditions  as 


FORMS  183 

to  mixing  and  laying  were  similar  to  those  of  the  first  tests  and  the  con- 
crete was  carried  up  to  a  height  of  10  feet  above  the  center  of  the  pressure 
face. 

"In  the  first  series  the  temperature  was  fairly  uniform  throughout, 
while  in  the  second  considerable  variation  was  experienced;  but  the 
effects  of  the  differences  in  temperature  on  the  lateral  pressure  cannot 
be  traced  and  would  appear  to  be  very  small. 

"The  general  conclusions  to  be  drawn  from  these  and  other  experi- 
ments is  that  the  lateral  pressure  of  concrete  for  average  conditions 
is  equivalent  to  that  of  a  fluid  weighing  85  pounds  per  cubic  foot.  *  *  * 
For  concrete  in  which  little  water  is  used  in  mixing,  the  pressure  is 
rather  less,  having  an  equivalent  fluid  value  as  low  as  70  Ibs.  per  cubic 
foot  in  very  dry  mixtures." 

There  is  apparently  a  large  divergence  of  pressures  as  experi- 
mentally obtained  and  until  more  extensive  experimentation  has 
been  performed  it  is  hardly  justifiable  to  use  other  than  an  empiric 
table  of  pressures;  guided,  however,  by  the  results  of  the  above 
quoted  work.  A  simple  code  may  be  used  as  indicated  below 
wherein  the  pressure  is  obtained  from  the  equation 

p  =  wh 

with  p  the  lateral  pressure  in  pounds  per  square  foot,  h  is  the 
concrete  head  in  feet,  and  w  is  to  be  used  as  follows : 

For  heights  of  concrete  less  than  5',  w  =  150 
For  concrete  5  to  10  feet,  w  =  100 

For  concrete  10  to  20  feet,  w  =    75 

For  concrete  over  20  feet,  w  =    50 

These  are  all  safe  values  and  insure,  when  used,  a  form  that  will 
not  yield. 

A  comparison  of  the  pressures  obtained  by  using  the  results  as 
tabulated  by  Major  Shunk  and  by  using  the  suggested  series  of 
values  just  given  show  quite  a  divergence  in  numerical  values. 
The  pressures  using  the  values  given  by  Major  Shunk  (the  curves 
giving  the  maximum  pressure  for  a  given  C  and  T  are  to  be  found 
on  p.  448,  "American  Civil  Engineers  Pocket  Book")  are  far 
lower  than  those  found  by  the  latter  method.  In  view  of  the 
fact,  however,  that  concrete  pressures  are  not  readily  formulated 
and  that  form  failures  have  demonstrated  that  such  pressures  do 
reach  a  high  value,  it  seems  better  to  follow  the  scheme  of  pres- 
sure intensities  given  above.  The  forms  should  be  designed 
then,  using  these  values  in  preference  to  using  the  experimental 
m  aximum  pressure. 


184 


RETAINING  WALLS 


Joist- 


The  extra  cost  of  the  stronger  forms  thus  obtained  is  far  less 
than  the  expense  entailed  in  remedying  the  result  of  a  form 
failure. 

At  the  end  of  the  chapter  a  problem  is  given  illustrating  the 
application  of  the  preceding  formulas  to  a  specific  example. 

Since  a  form  panel  may  be  placed 
at  any  point  of  the  face  of  the  wall, 
it  should  be  designed  for  the  maxi- 
mum pressure  that  can  come  upon 
it.  The  concrete  pressure  is 
carried  by  the  lagging  to  the  joists, 
which  in  turn  carry  it  to  the 
longitudinal  rangers.  These  carry 
the  load  to  tie  rods,  or  where  such 
rods  are  not  used,  to  shores  placed  against  the  rangers  (see 
Fig.  112). 

Lagging.  —  Generally  tongue  and  grooved  lumber  is  specified 
for  the  sheeting.  The  boards  are  continuous  over  the  joists  and 
with  the  support  of  the  tongue  and  grooving,  it  is  possible  to 
treat  the  panel  as  a  plate.  Ordinarily,  no  reliance  should  be 
placed  on  such  plate  action  and  the  boards  should  be  designed 
as  either  smple  or  fixed  beams.  Another  most  important  fea- 

TABLE  32.  —  SAFE  LOAD  PER  SQUARE  FOOT  ON  LAGGING 


FIG.  112.—  Typical  form  assembly, 


\ft 

L\ 

K%) 

'&> 

IH 

(if*) 

IK 

UK) 

2(1%) 

2>i(2H) 

2K(2^) 

2K(2«) 

3(2%) 

12 

1,000 

1,700 

2,500 

3,500 

4,700 

5,950 

7,500 

9,200 

11,000 

14 

750 

1,250 

1,850 

2,600 

3,450 

4,450 

5,550 

6,750 

8,100 

16 

600 

950 

1,400 

2,000 

2,650 

3,400 

4,250" 

5,200 

6,200 

18 

450 

750 

1,100 

1,550 

2,100 

2,650 

3,350 

4,100  , 

4,900 

20 

350 

600 

900 

1,250 

1,700 

2,150 

2,700 

3,300 

3,950 

22 

300 

500 

750 

1,050 

1,400 

1,800 

2,250 

2,750 

3,300 

24 

250 

400 

650 

900 

1,200 

1,500 

,900 

2,300 

2,750 

26 

200 

350 

550 

750 

1,000 

1,300 

,600 

2,000 

2,350 

28 

175 

300 

450 

650 

850 

1,100 

,400 

1,700 

2,050 

30 

160 

275 

400 

550 

750 

950 

,200 

1,500 

1,800 

33 

135 

225 

350 

450 

600 

800 

,000 

1,200 

1,500 

36 

110 

200 

300 

400 

500 

650 

850 

1,000 

1,200 

39 

100 

150 

250 

350 

450 

550 

700 

850 

1,050 

42 

85 

135 

200 

300 

400 

500 

600 

750 

900 

45 

75 

125 

175 

250 

350 

450 

550 

650 

800 

48 

65 

100 

160 

225 

300 

400 

450 

600 

700 

FORMS 


185 


TABLE  33. — SAFE  TIMBER  STRESSES  FOR  FORM  LUMBER 

(Taken  from  A.  R.  E.  A.,  railroad  timber  stresses,  the  stresses  increased 
50  per  cent,  because  of  the  nature  of  the  loading  and  the  temporary 
character  of  the  work.) 

Douglas  fir 1800 

Longleaf  pine 2000 

Shortleaf  pine 1600 

White  pine 1350 

Spruce 1500 

Norway  pine 1200 

Tamarack 1350 

Western  hemlock 1600 

Redwood 1350 

Bald  cypress 1350 

Red  cedar 1200 

White  oak..  1600 


TABLE  34. — SAFE  LOADS  ON  RANGERS  AND  JOISTS  IN  KIPS 


2'-0" 

3'-0" 

4'-0" 

^ 

2 

4 

6 

8 

10 

12 

2 

4 

6 

8 

10 

12 

2 

4 

6 

8 

10 

12 

2 

0.4 

0.9 

1.3 

1.8 

2.2 

2.7 

0.3 

0.6 

0.9 

1.2 

1.5 

1.8 

0.2 

0.4 

0.7 

0.9 

1.1 

1.3 

4 

1.8 

3.5 

5.3 

7.1 

8.8 

10.6 

1.2 

2.4 

3.6 

4.7 

5.9 

7.1 

0.9 

1.8 

2.7 

3.6 

4.4 

5.3 

6 

4.0 

8.0 

12.0 

16.0 

20.0 

24.0 

2.7 

5.3 

8.0 

10.7 

13.3 

16.0 

2.0 

4.0 

6.0 

8.0 

10.0 

12.0 

8 

7.1 

14.2 

21.2 

28.3 

35.4 

42.5 

4.7 

9.5 

14.2 

19.0 

23.6 

28.5 

3.6 

7.1 

10.7 

14.2 

17.8 

21.3 

10 

11.1 

22.1 

33.2 

44.4 

55.3 

66.2 

7.4 

14.8 

22.2 

29.6 

37.1 

44.5 

5.6 

11.1 

16.7 

22.2 

27.8 

33.3 

12 

16.0 

32.0 

48.0 

64.0 

80.0 

96.0 

10.7 

21.4 

32.0 

42.7 

53.4 

64.0 

8.0 

16.0 

24.0 

32.0 

40.0 

48.0 

5'-0" 

6'-0" 

7'-0" 

2 

0.2 

0.4 

0.5 

0.7 

0.9 

1.1 

0.1 

0.3 

0.4 

0.6 

0.7 

0.9 

0.1 

0.3 

0.4 

0.5 

0.6 

0.8 

4 

0.7 

1.4 

2.1 

2.8 

3.6 

4.3 

0.6 

1.2 

1.8 

2.4 

3.0 

3.6 

0.5 

1.0 

1.5 

2.0 

2.5 

3.0 

6 

1.6 

3.2 

4.8 

6.4 

8.0 

9.6 

1.3 

2.7 

4.0 

5.3 

6.7 

8.0 

1.1 

2.3 

3.4 

4.5 

5.7 

6.8 

8 

2.8 

5.7 

8.5 

11.4 

14.2 

17.0 

2.4 

4.7 

8.1 

9.5 

11.9 

14.2 

2.0 

4.1 

6.1 

8.1 

10.2 

12.2 

10 

4.4 

8.9 

13.4 

17.8 

22.2 

26.7 

3.7 

7.4 

11.1 

14.8 

18.7 

22.2 

3.2 

6.3 

9.5 

12.7 

15.8 

19.0 

12 

6.4 

12.8 

19.2 

25.6 

32.0 

38.4 

5.3 

10.7 

16.0 

21.3 

26.7 

32.0 

4.5 

9.1 

13.7 

18.3 

22.8 

27.4 

8'-0' 

lO'-O" 

2 

0.1 

0.2 

0.3 

0.4 

0.6 

0.7 

0.1 

0.2 

0.3 

0.4 

0.5 

0.6 

4 

0.4 

0.9 

1.3 

1.8 

2.2 

2.7 

0.4 

0.7 

1.1 

1.4 

1.8 

2.1 

6 

1.0 

2.0 

3.0 

4.0 

5.0 

6.0 

0.8 

1.6 

2.4 

3.2 

4.0 

4.8 

8 

1.8 

3.6 

5.3 

7.1 

8.9 

10.7 

1.4 

2.8 

4.3 

5.7 

7.1 

8.5                                .       ' 

10 

2.8 

5.6 

8.3 

11.1 

13.9 

16.7 

2.2 

4.4 

6.7 

8.9 

11.1 

13.3 

12 

4.0 

8.0 

12.0 

16.0 

20.0 

24.0 

3.2 

6.4 

9.6 

12.8 

16.0 

19.2 

186 


RETAINING  WALLS 


ture  is  the  amount  of  defection  permissible.     It  is  well  to  keep  the 
deflection  of  the  panel  within  one-eighth  of  an  inch. 

Table  32  gives  the  load  per  square  foot  to  be  carried  by  a  board 
12  inches  wide,  L  feet  long  (L  the  distance  between  joists)  and  h 
inches  thick.  The  unit  timber  stress  taken  is  1,000  pounds  per 
square  inch.  The  boards  are  designed  as  simple  beams.  Should 
the  permissible  stress  be  greater  than  that  used  here  the  load 
may  be  increased  in  direct  proportion  to  the  new  stress.  Again, 
if  the  board  is  to  be  treated  as  a  fixed  beam  the  load  to  be  carried 
may  be  increased  50  per  cent.  That  the  deflection  may  not 
exceed  one-eighth  of  one  inch,  for  simple  span. 

L  must  be  less  than  25  \/h 
and  for  a  fixed  span 

L  must  be  less  than  45  \/h 

Table  33  gives  a  range  of  unit  timber  stresses  for  several  woods. 

Table  34  gives  the  maximum  loads  to  be  carried  by  the  joists 
for  various  spacing.  The  thickness  of  the  joist  is  b  inches 
and  its  depth  h  inches.  The  loads  may  again  be  increased  in  the 
same  proportion  for  a  permissible  unit  stress  greater  than  one 
thousand  pounds  per  square  inch  and  again  when  the  beam  is 
assumed  as  fixed  in  place  of  simply  supported.  This  same  table 
may  also  be  used  to  design  the  rangers  supporting  the  panels. 

Tie-rods. — The  diameter  of  the  tie-rod  depends  upon  the  size 
of  the  panel  supported  and  its  position  in  the  form.  The  con- 
crete pressures  may  be  taken  from  the  empiric  scheme  given  on 

TABLE  35. — LOADS  IN  LBS.  ON  TIB  RODS 


Rod  diam- 
eter 

±  (irmisBiu.it:  umc  stresses 

12,000 

16,000 

20,000 

25,000 

H 

150 

200 

250 

300 

H 

600 

800 

1,000 

1,200 

H 

1,320 

1,750 

2,200 

2,700* 

y* 

2,350 

3,150 

4,000 

4,900 

% 

3,700 

4,900 

6,100 

7,700 

H 

5,300 

7,100 

8,800 

11,000 

% 

7,200 

9,650 

12,000 

15,000 

i 

9,400 

12,700 

15,700 

19,600 

m 

14,700 

19,700 

24,500 

30,700 

FORMS  187 

page  183.  The  unit  stress  in  the  steel  is  usually  taken  at  16,000 
Ib.  per  square  inch.  Small  diameter  rods  may  be  pulled  out  and 
this  should  be  borne  in  mind  in  selecting  the  rod  spacing.  Table 
35  gives  the  load  on  tie  rods  for  a  range  of  unit  steel  stresses. 

A  simple  detail  carrying  the  tie  rod  load  is  shown  in  Fig.  112. 
This  obviates  the  necessity  of  boring  a  large  timber  to  allow  the 
rod  to  pass  through.  The  tie  rods  may  be  threaded  on  the 
end  and  fastened  to  the  rangers  by  nuts  and  washers,  or  a  patented 
support,  such  as  the  universal  clamp  (Universal  Clamp  Co.) 
may  be  used  on  a  plain  round  bar. 

Rangers.  —  The  rangers  themselves  may  be  designed  as  simple 
or  fixed  beams,  with  spans  between  tie  rods  and  carrying  the 
joists.  If  the  ranger  is  to  be  b  inches  wide  and  h  inches  deep, 
with  span  between  tie  rods  L,  then 


WL/I  =  pbhz/Q  and  bh*  =  (184) 

7  may  be  taken  as  8  or  12,  depending  upon  the  assumption  that 
the  beam  is  a  fixed  or  simple  one;  and  p  may  be  taken  as  the  safe 
permissible  unit  stress  in  the  timber. 

Form  Re-use.  —  If  the  panels  are  built  in  stout  units,  carefully 
put  together,  they  may  be  used  several  times.  When  the  lagging 
becomes  splintered  marring  the  face  of  the  concrete  and  making  it 
very  difficult  to  strip  the  form,  the  form  should  be  abandoned. 
With  care  in  placing  and  stripping  the  forms,  a  panel  maybe  used 
from  3  to  10  times.  Two  inch  tongue  and  grooved  sheeting  makes 
a  good  strong  form  but  its  weight  limits  it  to  small  panels.  If 
plant  is  available  to  handle  these  units,  this  objection  is  removed 
and  as  large  sections  maybe  used  as  is  found  convenient  to  assem- 
ble. Usually  a  limiting  section  would  be  about  8'  by  10'. 

Form  Work.  —  It  is  essential  that  a  careful  study  be  made  of  the 
form  work,  taking  into  consideration  the  expected  daily  output 
of  concrete  and  the  time  the  forms  must  remain  in  place.  It  must 
be  remembered  that  forms  of  simple  shape,  quickly  assembled, 
put  in  place  and  stripped,  make  for  large  economy  on  the  work. 
Skilled  carpenters  will  prepare  excellent,  well-fitting  forms  of  long 
duration.  It  is  a  poor  economy  to  substitute  for  such  labor  the 
ordinary  wood  butcher,  a  most  competent  man  in  his  sphere. 

In  this  connection  the  use  of  a  portable  machine  saw,  propelled 
electrically  or  by  gasoline  is  a  marvellous  labor  and  time  saver 
and  few  jobs,  however  small,  can  afford  to  be  without  one. 


188  RETAINING  WALLS 

The  rear  and  face  forms  of  the  wall  are  kept  the  proper  dis- 
tance from  each  other  by  means  of  wooden  separators  called 
spreaders.  When  the  tie  rods  are  placed  or  wire  used  in  place  of 
tie  rods  and  tension  put  on  them  the  spreaders  are  held  in  place 
without  any  further  details.  As  the  concrete  is  poured  and 
reaches  the  lever  of  a  spreader,  the  spreader  is  knocked  out.  The 
tie  rods  and  wires  must  remain  until  the  concrete  has  set  (see 
later  chapter) . 

Bracing. — Bracing,  or  shoring  is  necessary  to  take  care  of 
unbalanced  pressures  and  the  possible  overturning  of  the  form 
due  to  the  vibrations  and  shocks  set  up  during  the  pouring  of  the 
concrete.  Such  stresses  are  obviously  not  to  be  computed  and 
experience  alone  dictates  the  proper  amount  of  bracing  to  be  used. 
They  are  made  usually  of  4  inch  by  4  inch,  or  4  in.  by  6  in.  stock, 
nailed  to  the  rangers  and  held  against  foot  blocks  or  stakes  in  the 
ground  (see  Fig.  113).  Where  concrete  is  to  be  poured  against 
a  permanent  mass,  requiring  forms  on  one  side  only,  no  tie  rods 
or  wires  can  be  used  through  the  concrete  and  the  bracing  on 
the  one  side  must  take  the  full  concrete  pressure  and  are  to  be 
designed  accordingly.  When  walls  of  some  height  are  to  be 
poured  in  several  lifts,  an  overlapping  of  the  joists  may  render 
bracing  unnecessary  above  the  lower  lift. 


FIG.  113. — Form  FIG.  114. — Holding  forms 

brace.  by  bolt  in*concrete. 

Occasionally  the  environment  is  such  that  bracing  cannot  be 
used  on  either  side.  It  is  possible  here  to  concrete  eye  bolts 
into  the  bottom  lift  and  into  each  succeeding  lift  and  to  anchor 
the  forms  to  these  (see  Fig.  114). 

Generally  an  excessive  amount  of  bracing  is  used,  with  a  result- 
ing forest  of  timber  and  making  it  impossible  to  run  plant  close 
to  the  forms.  Form-work  is  a  fertile  field  of  study  for  the  engineer 
and  the  designing  and  detailing  of  such  work  is  worthy  of  as 
serious  attention  as  the  design  and  construction  of  the  wall  itself. 

Stripping  Forms. — It  is  essential  that  forms  be  stripped  as 
soon  as  it  is  possible  to  do  so.  To  keep  a  form  in  place  longer 
than  is  required  makes  it  impossible  to  get  the  full  economical 


FORMS  189 

reuse  of  the  form  and  makes  it  very  difficult  to  finish  and  repair 
the  concrete  surface.  In  the  warm  summer  months  the  forms 
may  be  stripped  after  24  hours.  In  the  spring  and  fall  months 
they  should  be  left  in  place  from  48  to  72  hours.  When  in  doubt 
as  to  the  hardness  of  the  concrete  a  small  portion  of  the  form 
may  be  taken  off  and  a  thumbnail  impression  made.  If  there 
is  no  indentation,  it  is  safe  to  take  off  the  balance  of  the  forms. 
If  it  is  possible  to  remove  the  tie-rods  (rods  J£  inch  or  less  in 
diameter  may  be  economically  recovered;  rods  of  larger  diameter 
are  usually  left  in  the  wall)  these  should  be  taken  out  before  the 
forms  are  stripped.  Patent  rod  pullers1  may  be  used  to  take 
out  the  rods.  Where  the  rods  are  left  in  the  wall,  they  should 
be  cut  back  an  inch  to  an  inch  and  a  half  and  the  face  of  the  wall 
plastered  at  these  points.  Wires  are  rarely  recovered  and  are 
cut  off  in  the  same  fashion  as  the  rods.  The  sooner  after  strip- 
ping these  rods  and  wires  are  cut,  the  easier  it  is  to  repair  and 
finish  the  face  of  the  wall  (see  later  chapter  of  wall  finish). 

From  ten  days  to  two  weeks  of  favorable,  warm  weather 
should  elapse  before  the  fill  is  placed  behind  the  wall.  If  the 
fill  is  to  be  placed  at  a  rapid  rate,  e.g.,  by  dump  cars  from  a  tem- 
porary trestle  and  the  like,  a  greater  period  of  time  should  elapse. 
This  is  especially  important  for  the  reinforced  concrete  walls, 
where  the  concrete  will  receive  the  full  load  immediately  after 
the  completion  of  the  embankment. 

Oiling  and  Wetting  Forms. — A  dry  form  will  absorb  the  water 
from  the  concrete,  in  the  process  of  curing,  leaving  a  peculiar 
pock-marked  appearance  of  the  concrete  face  due  to  the  honey- 
combing of  the  surface.  The  forms  should  be  wetted  by  pail 
or  hose  immediately  before  the  concrete  pour  is  started.  To  aid 
in  the  stripping  of  the  form,  the  inside  face  of  the  form  is  usually 
oiled,  with  a  heavy  oil,  termed  a  form  oil,  which  is  a  heavy 
sludge.  Although  this  stains  the  concrete  face,  the  rubbing 
and  washing  of  the  concrete  surface  easily  removes  the  oil  marks. 

Patent  Forms. — For  a  wall  of  large  yardage  and  of  fairly 
constant  outline,  permitting  many  reuses  of  the  form  panel,  the 
use  of  some  of  the  patent  forms  may  show  quite  an  economy, 
both  in  the  construction  of  the  form  and  in  the  labor  of  setting 
up  and  stripping  the  forms. 

The  two  best  known  types  of  such  forms  are  the  Hydraulic 
Pressed  Steel  Form  and  the  Blaw  Form. 

1  An  excellent  rod  puller  is  sold  by  the  Universal  Clamp  Co.  of  Chicago. 


190 


RETAINING  WALLS 


The  Hydraulic  Pressed  Steel  Form  consists  of  two  parts:  the 
bracing  and  the  form  panel.  The  bracing  is  formed  of  upright 
Us  spliced  as  necessary  and  held  together  by  tie  rods  and  spacers 
or  liners.  Fig.  115  shows  a  sketch  of  the  bracing  and  its  details. 


•Liners  Punched 'f  "Centers 
forAdjustment  of 
\      Uprights-^ 


Wood  Separator 
not  Furnished 
nrfh  Forms 


TZa>'*^~- «^3# 

FIG.  115. —  Hydraulic  Pressed  Steel  Co.  form  assembly  of  liners  and  plates. 

The  form  panel  consists  of  a  sheet  metal  (all  metal  used  in  these 
forms  both  panel  and  uprights  are  no.  11  gage,  i.e.,  one-eighth  inch 
metal)  backed  by  1"  boards.  Around  the  periphery  of  the  panel 
a  U  steel  edge  is  put,  to  which  the  boards  are  screwed  (see  Fig. 
116).  The  panels  are  held  in  place  against  the  uprights  by 
means  of  stout  Us  spaced  about  one  foot  apart  (see  Fig.  115). 

•  Standard  Upright 


Standard"' 
Wall  Plate 


Wectge.._ 


'-  Standard 
Wall  Plate 
Standard  Wall 
Plate  Clamp. 


FIG.  116. — Section  of  Hydraulic  Pressed  Steel  Co.  form. 

It  is  claimed  by  the  company  that  the  panels  may  be  reused 
about  300  times  before  wearing  out.  Where  a  job  will  permit  a 
reuse  of  the  form  panel  exceeding  twenty  or  thirty  times,  they 
maintain  that  their  form  will  prove  cheaper  than  the  wood  form 
ordinarily  built. 


PLATE  IV 


a  page  191) 


FORMS 


191 


The  advantages  of  the  form  are  quite  obvious.  The  uprights 
may  be  built  up  to  the  top  of  the  wall.  After  the  lower  lift  of 
the  wall  is  poured  no  further  bracing  becomes  necessary,  since 
the  form  is  now  anchored  against  the  lower  half  of  the  wall. 
The  panels  may  be  removed  after  twenty-four  hours,  the  uprights 
and  liners  remaining  as  much  longer  as  is  necessary  before  the 
wall  is  self-supporting. 

The  panels  need  only  be  put  in  as  the  concrete  comes  near 
their  level,  thus  permitting  a  thorough  spading  and  tamping  of 
the  mass:  quite  a  vital  point  where  the  wall  is  thin  or  has  a 
special  shape. 

Blawform. — The  Blawform  consists,  essentially  of  a  steel 
panel,  reinforced  with  angle  on  the  back  and  held  in  place  with 
a  steel  assembly  of  joists  and  rangers.  By  an  ingenious  travel- 
ling gantry  device,  the  form  panels  are  braced  against  this  travel- 
ler, which  runs  on  rails  alongside  the  work.  A  large  number  of 
instances  of  their  use  for  both  heavy  and  light  retaining  walls 
are  given  in  their  Catalogue  16. 

Supporting  the  Rod  Reinforcement. — Since  most  of  the  rod 
system  in  a  reinforced  concrete  wall  must  be  in  place  before 
the  concrete  pouring  is  started,  some  means  of  support  must  be 
provided.  In  the  "L"  or  "T" 
shaped  cantilevers,  the  heavy 
rod  system  of  the  vertical  arm 
extends  into  the  footing  and 
must,  therefore,  be  set  up  and 
in  place  before  the  wall  forms 
are  up.  Many  simple  devices 
may  be  'used  for  this  purpose. 
Fig.  117,  (See  Fig.  B,  Plate 
IV)  shows  a  typical  and  effi- 
cient method  of  taking  care  of  these  rods.  When  the  footing 
has  been  poured,  thereby  anchoring  these  rods,  the  wall  forms 
are  set  in  place  and  the  rods  are  wired  and  held  the  required 
distance  away  from  the  concrete  face.  The  horizontal  rod 
system  is  wired  to  the  vertical  rods  and  helps  to  maintain  the 
proper  spacing  of  these  vertical  rods.  Patent  wire  clips  may 
be  used  to  wire  the  horizontal  and  vertical  rods  together. 

The  horizontal  rods  in  the  footing,  itself,  are  laid  in  the  con- 
crete when  the  proper  level  has  been  reached.  It  is  preferable  to 
wire  a  net  of  these  rods  together  before  placing  in  the  wet  con- 


Stones  as 

Counterweights .. 


Mr":  Longitudinal 
I   Rods 


.-•3x  12"  by  10',6on 
Centers 


FIG.  117. — Supporting  rod  reinforce- 
ment of  cantilever  wall. 


192  RETAINING  WALLS 

crete  to  make  sure  that  the  proper  spacing  as  called  for  on  the 
plans,  will  be  kept. 

The  rod  systems  of  the  other  types  of  the  reinforced  concrete 
walls  are  supported  and  placed  in  similar  fashion.  The  problem 
of  supporting  the  rods  extending  into  the  footing  for  the  slab 
types  of  wall  is  comparatively  a  simple  one,  since  these  rods  are 
the  light  system  and  therefore  need  little  framework  to  carry 
them.  The  main  system  (particular  stress  is  placed  upon  the 
counterfort  and  box  types  of  wall)  is  suspended  to  the  forms  in 
the  usual  manner  and  kept  at  the  proper  distance  away  from  the 
face  by  means  of  small  wooden  spreaders  which  are  removed  in 
pouring  as  quickly  as  the  concrete  reaches  their  level.  The  tie 
rods  form  a  good  support  for  the  horizontal  rods  and  are  generally 
so  used. 

It  is  important  that,  whatever  method  of  support  is  employed, 
the  rods  should  be  held  firmly  in  place.  Spading  and  spouting 
of  concrete  are  liable  to  shift  the  rods  unless  they  are  stoutly 
supported.  It  is  understood  that  in  the  design  of  walls  involving 
intricate  rod  systems  (see  Chapter  4)  proper  consideration  has 
been  given  to  the  practicability  of  the  rod  layout  and  to  the 
feasibility  of  supporting  the  rods  and  of  pouring  the  concrete. 
Simplicity  of  rod  design  insures  an  easy  concrete  pour  and  leaves 
the  engineer  with  a  reasonable  assurance  that  the  rods  are  finally 
placed  where  they  were  originally  designed  to  go. 

The  rod  system  has,  presumably,  been  carefully  and  economi- 
ally  designed  and  no  variations  in  spacing  should  be  permitted 
in  the  field,   except  in  isolated  instances,  where  a 
proper  attempt  should  then  be  made  to  reinforce  the 
weak  spots  resulting.     Leaving  openings  in  the  walls 
for  construction  reasons,  as,  for  example,  to  permit 
placing  timbers  through  the  wall,  or  to  place  large 
FIG'  us     P*Pe  e^c>>  w^  result,  when  the  wall  is  finally  patched 
in  portions  being  without  the  proper  reinforcement. 
The  rods  should  be  bent  around  these  openings  as  shown  in 
Fig.  118. 

Undoubtedly  walls  are  at  times  designed  with  excessive  rein- 
forcement due  to  indifference  or  carelessness  and  the  knowledge 
of  such  excessive  strength  has  encouraged  the  engineers  in  the 
field  and  the  contractors  constructing  such  walls  to  alter  the 
rod  spacing  to  accommodate  minor  construction  exigencies. 
Such  acts  are,  in  the  main,  unfortunate  and  designs  which  can 


FORMS  193 

safely  permit  many  such  liberties  are  to  be  deplored.  Walls 
should  be  designed  as  economically  as  possible  with  due  considera- 
tion for  all  contingencies  and  when  a  design  has  left  the  hands 
of  a  competent,  conscientious  engineer,  no  changes  should  be 
permitted  in  the  field  save  with  the  concurrence  of  the  man 
responsible  for  the  design. 

Travelling  Forms. — Engineering  News,  Vol.  73,  p.  67.  Track 
Elevation  Rock  Island  Lines  Chicago. 

"The  walls  are  built  in  travelling  forms  which  straddle  the  site  of  the 
wall  and  are  carried  by  wheels  on  either  side.  Both  wood  and  steel 
forms  of  this  type  are  used,  each  being  long  enough  for  a  35  foot  section 
and  having  grooved  wheels  riding  upon  two  lines  of  rails.  *  : 
The  abutments  are  built  in  fixed  forms  of  the  usual  type.  Plank 
sheeting  is  used  in  both  cases  and  the  two  lines  of  sheeting  are  held 
together  by  tie-rods  instead  of  wires.  The  rods  are  plain  bars,  not 
threaded,  and  are  fitted  with  clamps  instead  of  nuts.  When  a  clamp 
is  in  place,  a  set  screw  jams  the  rod  against  a  V  slot  in  the  clamp,  securing 
it  rigidly  in  position.  (Engineering  News,  Sept.  10,  1914).  Each 
rod  is  imbedded  in  a  tin  tube,  so  that  it  can  be  withdrawn  readily, 
the  holes  being  then  packed  with  stiff  cement  grout  at  each  end. 

"The  retaining  walls  are  built  in  alternate  sections  of  35  feet  with 
the  travelling  forms.  It  takes  about  six  hours  to  fill  the  form,  which 
is  then  left  in  place  about  15  hours.  It  then  takes  about  20  hours  to 
release  the  travelling  form  move  seventy  feet  forward  and  adjust  them 
and  the  sheeting  ready  for  the  concrete.  The  use  of  the  travelling  forms 
has  enabled  the  work  to  be  done  in  about  25  per  cent,  of  the  time  re- 
quired with  the  ordinary  forms  (from  the  building  to  the  removal  of 
the  form)  and  at  about  50  per  cent,  of  the  cost  (including  erecting,  pour- 
ing and  dismantling)." 

New  Passenger  Terminal,  C.  &  N.  W.  R.  R.  Armstrong, 
Journal  of  the  Western  Society  of  Engineers,  Vol.  16. 

"  The  forms  were  built  in  sections  30  feet  long.  The  footings  were 
first  built  and  allowed  to  set.  The  forms  for  the  super  walls  were  then 
built.  It  was  required  that  an  entire  section  of  superwall  should  be 
poured  at  one  continuous  run  of  the  mixing  plant,  in  order  that  no  hori- 
zontal joints  might  occur  in  the  walls.  The  forms  were  constructed 
of  4-inch  by  6-inch  studding  and  2-inch  by  8-inch  dressed  and  matched 
sheeting.  The  two  sides  were  tied  together  with  %-inch  rods  which 
were  passed  through  iron  pipes  consisting  of  old  boiler  flues.  The  rods 
were  drawn  out  when  the  forms  were  removed,  but  the  pipes  were  left 
in  place,  the  opening  in  the  face  of  the  wall  being  filled  with  mortar." 

13 


194  RETAINING  WALLS 

Forms  Built  in  Central  Yard. — Engineering  and  Contracting, 
June  11,  1913,  p.  649.  Track  Elevation,  Chicago,  Milwaukee 
and  St.  Paul  R.  R.  For  this  work  the  forms  were  built  in  a 
central  yard  and  were  shipped  out  to  the  work  as  required  on 
flat  cars.  They  were  taken  from  the  cars  and  set  in  place  by 
means  of  locomotive  cranes. 

Erecting  Forms  on  Curves. — R.  H.  Brown,  Engineering 
Record,  Vol.  61,  p.  714. 

"There  is  nothing  more  unsightly  in  concrete  work  than  to  see  the 
impression  of  the  forms  running  out  of  level.  A  great  deal  of  pains 
is  taken  to  produce  smooth  surfaces  by  spading,  but  very  little  attention 
is  given  to  the  mold  itself.  This  is  very  noticeable  in  massive  work.  On 
a  straight  wall  there  is  no  excuse  for  this,  but  in  building  forms  on  curves 
of  short  radius  there  is  great  difficulty  in  making  a  symmetrical  sur- 
face and  eliminating  the  segmental  effect.  If  the  following  method  is 
carried  out  a  piece  of  concrete  will  be  produced  which  is  a  true  curve 
in  every  foot  of  its  length. 

"Take  a  wire  about  the  size  of  that  used  in  telephone  lines  and  upon 
a  smooth  level  surface  strike  on  the  board  an  arc  of  the  radius  of  the 
center-line  of  the  wall  or  dam.  Arc  of  radius  of  150  feet  can  easily  be 
handled.  Care  must  be  used  in  doing  this  that  the  wire  is  always 
straight.  This  template  is  now  sawed  out  on  a  band  saw  in  about  ten- 
foot  lengths.  The  rear  and  face  templates  can  be  struck  from  this  one 
by  means  of  a  T-square. 

"Run  out  the  center  line  of  the  wall  in  chords  of  10  feet  and  put  in 
permanent  plugs  at  these  points.  Erect  a  well-braced  series  of  batters 
around  the  curve  and  set  the  top  ledger  board  at  the  exact  crest  of  the 
wall.  Place  the  center-line  templates  on  these  boards  and  plumb  them 
over  the  plugs,  cleating  them  together  as  fast  as  they  are  put  in  correct 
position.  With  this  center  to  work  from,  the  outside  and  inside  curves 
can  be  set. 

"Make  two  boards  four  feet  long,  one  edge  straight  and  the  other 
bevelled  to  the  batter  of  the  front  and  rear  faces  respectively.  The 
studding  can  now  be  set,  using  a  carpenter  level.  The  upper  end  will 
rest  against  the  template,  the  lower  end  following  the  inequalities  of  the 
ground. 

"Start  the  bottom  board  as  low  as  possible  and  run  it  along  the 
curve  on  both  sides  making  it  absolutely  level.  The  rest  of  the  board- 
ing can  now  be  nailed  to  the  studding,  springing  each  one  carefully 
into  place.  The  purlins  (Rangers)  are  put  in  and  rods  run  through 
and  tightened.  After  everything  is  well  braced,  remove  the  batter 
boards  used  in  lining  up.  When  the  forms  are  removed  a  true  curve 
is  presented  to  the  eye." 


FORMS 


195 


Problems 

It  is  required  to  design  and  construct  a  set  of  forms  for  a  wall  30  feet  high 
above  the  footing  with  expansion  joints  40  feet  apart,  of  section  shown  in 
Fig.  119.  It  is  figured  that  the  mixer  can  pour  100  yards  of  concrete  in  an 
8-hour  shift,  this  to  govern  the  lift  of  concrete  poured. 

The  portion  of  the  wall  requiring  forms  contains  a  volume  between  ex- 
pansion joints  of  93  cubic  yards.  It  is  thus  possible  to  complete  the  pouring 
of  the  section  in  one  continuous  pour  within  the  time  specified — the  ideal 
arrangement.  The  forms  will  be  designed  upon  this  basis. 


'0.70k 


1.33k 
'1.5k 
-50/fys 

Reduced  Loading  ^K( 
Diagram 


Spacing  of  Rangers 


FIG.  119. 


Concrete  Pressures. — On  the  basis  of  Major  Shunk's  experiments,  the 
concrete  pressure  at  the  base  is  determined  as  follows :  (It  is  assumed  that 
the  concrete  enters  the  form  at  the  temperature  of  70°.)  Since  the  con- 
crete form  is  30  feet  high  and  is  filled  in  8  hours,  the  rate  of  filling  per  hour 
is  3.75  feet,  the  value  of  R  to  be  used  in  the  work  following.  From  Table  31 
with  the  temperature  of  70°,  c  =  25  and  from  (183) 

T  =  25  +  150/3.75  =  65  minutes  =  1.1  hrs. 

The  maximum  pressure  that  can  occur  is  found  by  employing  the  curves  of 
Major  Shunk,  which  can  be  found  in  the  American  Civil  Engineers'  Pocket 
Book,  page  448.  This  maximum  pressure,  with  the  value  of  c  and  T  as 
above  found  is  850  pounds  per  square  foot.  Using  the  empiric  rule  given  on 
page  183,  the  pressure  tunction  to  use  is  50  Ib.  per  square  foot,  which  would 
give,  at  the  base  of  the  wall  30  X  50  or  1500  Ib.  per  square  foot.  The  average 
pressures  found  by  Robinson,  page  182,  of  85  Ib.  per  square  foot  intensity 
would  give  a  base  pressure  of  85  X  30  =  2550  Ibs.  far  in  excess  of  both  of  the 
pressures  just  found.  The  experimental  value  of  85  Ibs.  is  based  upon  heads 
not  exceeding  10  feet — and  is  therefore  of  little  application  to  the  case  at 
hand.  Again,  the  experiments  of  Major  Shunk,  while  most  admirably  and 
extensively  performed  cannot  be  made  the  final  basis  for  concrete  pressure 
determination.  It  is  therefore  logical  to  employ,  awaiting  more  experi- 
mental data,  the  empiric  table  suggested  in  the  previous  pages  and  the  form 
work  of  the  given  problem  will  be  designed  upon  the  table  quoted. 


196  RETAINING  WALLS 

In  line  with  the  recommendations  of  the  text,  2-inch  tongue  and  grooved 
sheeting  will  be  used.  North  Carolina  spruce  dressed  all  sides  will  permit 
a  working  stress  for  the  form  work  of  1500  Ibs.  per  square  inch.  The  sheeting 
will  be  treated  as  continuous,  so  that  the  product  kp  of  Table  32  is  1500  X 
12  =  18000.  Since  the  loads  on  the  sheeting  of  Table  32  employ  the  con- 
stant 8000,  to  use  the  table  directly,  the  above  load  of  1500  pounds  per 
square  inch  will  be  reduced  in  the  ratio  of  1800%ooo>  °r  will  become  670 
Ib.  per  square  inch.  For  2"  material,  the  dressed  thickness  is  1%"  and 
the  table  shows  that  a  load  of  670  pounds  will  permit  the  joists  to  be  spaced 
30  inches  apart.  In  view  of  the  fact  that  the  forms  are  to  be  used  several 
times,  the  panels  may  be  set  at  any  position  in  the  form,  and  will  therefore 
all  be  constructed  alike,  and  of  the  heaviest  dimensions  required. 

The  rangers  are  set  after  the  panels  are  in  place  and  may  therefore  be 
spaced  to  accommodate  the  concrete  pressures.  A  good  working  size  for  a 
joist  is  a  4-inch  by  6-inch  stick.  Fig.  119  gives  the  load  layout  for  the  30- 
inch  spacing  of  the  joists.  The  loads  have  been  divided  by  the  constant 
2.25  i.e.,  the  ratio  of  1800%ooo>  to  permit  a  direct  use  of  the  Tables. 
Table  34  is  to  be  used  in  the  design  of  the  joists.  Let  the  lower  ranger  carry 
a  three-foot  panel  of  sheeting.  From  the  figure,  the  lower  three  feet  bring 
a  tabular  equivalent  load  of  4.8  kips.  Table  34  permits  a  three-foot  spacing 
of  this  size  joist  and  accordingly  the  first  ranger  having  been  placed  as  close 
to  the  bottom  as  is  feasible,  the  next  will  be  spaced  three  feet  above  it.  A 
similar  study  of  the  loading  above  the  lower  panel  shows  that,  to  maintain 
the  same  size  of  joist,  the  next  four  rangers  must  be  spaced  on  three  feet 
centers.  The  remainder  of  the  spacing  is  shown  on  Fig.  119. 

The  rangers  will  be  made  up  of  two  3-inch  by  6-inch  sticks,  a  handy  mer- 
chantable size.  The  safe  load  span  upon  these  two  pieces  will  determine 
the  tie-rod  spacing.  From  equation  (184)  page  187,  with  7  =  12;  p  = 
1500  as  before,  b  =  6  and  h  =  6, 

WL  =  648,000 

or  if  w  is  the  load  per  linear  foot  upon  the  ranger  and  L  is  the  length  expressed 
in  feet 

wL*  =  54,000 

The  lower  ranger  will  carry  4500  Ibs.  per  linear  foot  (the  actual  loads  are 
used  here),  whence 

L  =  3'  6" 

The  tie  rods  will  accordingly  be  spaced  3'  6"  apart  at  the  lower  lift  of  rangers. 
The  panel  load  that  a  tie-rod  will  be  called  upon  to  carry  is 

3.5  X  3  X  1500  =  15,700  Ib. 

To  avoid  using  large  size  tie-rods,  which  cannot  be  recovered  two  tie  rods 
will  be  used  together  at  the  lower  lift.  From  Table  35  with  a  unit  stress  of 
16,000  pounds  per  square  inch  for  steel,  two  %-inch  rods  will  be  used. 
The  other  tie  rod  spacing,  and  the  necessary  rod  section  are  both  found 
by  identical  means. 


CHAPTER  V11I 
CONCRETE  CONSTRUCTION 

Water  Content. — Recent  years  have  noted  a  marked  increase 
in  the  knowledge  of  the  proper  mode  of  selecting  and  mixing 
the  aggregates  necessary  to  produce  good,  strong  concrete 
masonry.  Not  only  must  the  various  aggregates  be  put  in  the 
correct  proportions,  but  the  amount  of  water  used  is  vitally 
important.  The  excess  or  deficiency  of  water  seriously  affects 
the  strength  of  the  concrete. 

Each  element  entering  into  a  concrete  mix  performs  a  definite 
and  separate  function  and  each  is,  accordingly,  capable  of  affect- 
ing favorably  or  unfavorably  the  strength  of  the  concrete. 
Concrete  is  usually  so  proportioned  that  each  finer  material  fills, 
more  or  less  completely,  the  voids  in  the  coarser  aggregate  (see 
following  pages  on  Prof.  Abrams  demonstration  that  the  strength 
of  the  concrete  does  not  require,  prima  facie,  this  condition). 
The  action  of  water  is  in  part  a  solvent  and  in  part  a  chemical  one. 
The  results  of  Mr.  Nathan  C.  Johnson1  and  other  laboratory 
investigators  have  strikingly  demonstrated  the  vital  importance 
of  the  correct  amount  of  water  and  it  has  been  shown  that  con- 
crete failures,  both  partial  and  complete  are  attributable  to  excess 
of  water.  The  evaporation  of  this  excess  amount  of  water  leaves 
pockets  and  crevices  in  the  concrete,  materially  reducing  the 
effective  area  capable  of  resisting  stress.  The  widely  varying 
results  of  concrete  tests  and  the  necessary  high  factors  of  safety 
are  thus  quite  obviously  explained. 

Prof.  Talbot2  has  made  a  series  of  timely  pointers  on  concrete, 
some  of  which  may,  with  profit,  be  quoted  here. 

''The  cement  and  the  mixing  water  may  be  considered  together  to 
form  a  paste;  this  paste  becomes  the  glue  which  holds  the  particles  of 
the  aggregate  together. 

1  Engineering  News  Record,  June26, 1919,  p.  1266.     Also  "Better  Concrete— 
The  Problem  and  Its  Solution,"  N.  C.  JOHNSON,  Journal  Engineer's  Club, 
Philadelphia,  Pa. 

2  Engineering  News-Record,  May  1,  1919  for  a  resume^  of  his  remarks  at 
the  annual  convention  of  the  American  Railway  Engineering  Association. 

197 


198  RETAINING  WALLS 

"  The  volume  of  the  paste  is  approximately  equal  to  the  sum  of  the 
volume  of  the  particles  of  the  cement  and  the  volume  of  the  mixing 
water. 

"The  strength  given  this  paste  is  dependent  upon  its  concentration — 
the  more  dilute  the  paste  the  lower  its  strength;  the  less  dilute  the  greater 
its  strength. 

"The  paste  coats  or  covers  the  particles  of  aggregate  partially  or 
wholly  and  also  goes  to  fill  the  voids  of  the  aggregate  partially  or  wholly. 
Full  coating  of  the  surface  and  complete  filling  of  the  voids  are  not 
usually  obtained. 

"The  coating  or  layer  of  paste  over  the  particles  forms  the  lubricating 
material  which  makes  the  mass  workable;  that  is,  makes  it  mobile  and 
easily  placed  to  fill  a  space  compactly. 

"The  requisite  mobility  and  plasticity  is  obtained  only  when  there 
is  sufficient  paste  to  give  a  thickness  of  film  or  layer  of  paste  over  the 
surface  of  the  particles  of  aggregate  and  between  the  particles  sufficient 
to  lubricate  these  particles. 

"Increase  in  mobility  may  be  obtained  by  increasing  the  thickness 
of  the  layer  of  paste;  this  may  be  accomplished  either  by  adding  water 
(resulting  in  a  weaker  paste)  or  by  adding  cement  up  to  a  certain  point 
(resulting  in  a  stronger  paste) . 

"Factors  contributing  to  the  strength  of  concrete  are  then,  the  amount 
of  cement,  the  amount  of  mixing  water,  the  amount  of  voids  in  the 
combination  of  fine  and  coarse  aggregate  and  the  area  of  surface  of  the 
aggregate. 

"For  a  given  kind  of  aggregate  the  strength  of  the  concrete  is  largely 
dependent  upon  the  strength  of  the  concrete  paste  used  in  the  mix, 
which  forms  the  gluing  material  between  the  particles  of  the  aggregate. 

"For  the  same  amount  of  cement  and  same  voids  in  the  aggregate, 
that  aggregate  (or  combination  of  fine  and  coarse  aggregates)  will  give 
the  higher  strength  which  has  the  smaller  total  area  of  surface  of  par- 
ticles, since  it  will  require  the  less  amount  of  paste  to  produce  the  re- 
quisite mobility  and  this  amount  of  paste  will  be  secured  with  a  smaller 
quantity  of  water;  this  paste  being  less  dilute  will  therefore  be  stronger. 
The  relative  surface  area  of  different  aggregates  or  combination  of 
aggregates  may  readily  be  obtained  by  means  of  a  surface  modulus 
calculated  from  the  screen  analysis  of  the  aggregate. 

"For  the  same  amount  of  cement  and  the  same  surface  of  aggregate, 
that  aggregate  will  give  the  higher  strength  which  has  the  less  voids, 
since  additional  pore  space  will  require  a  larger  quantity  of  paste  and 
therefore  more  dilute  paste. 

"Any  element  which  carries  with  it  a  dilution  of  the  cement  paste 
may  in  general  be  expected  to  weaken  the  concrete.  Smaller  amounts 
of  cement,  the  use  of  additional  mixing  water  to  secure  increased  mo- 


CONCRETE  CONSTRUCTION  199 

bility  in  the  mass,  increased  surface  of  aggregate,  and  increased  voids 
in  the  aggregate  all  operate  to  lower  the  strength  of  the  product. 

"In  varying  the  gradation  of  aggregate  a  point  will  be  reached,  how- 
ever, when  the  advantage  in  the  reduction  of  surface  of  particles  is  offset 
by  increased  difficulty  in  securing  a  mobile  mass,  the  voids  are  greatly 
increased,  the  mix  is  not  workable  and  less  strength  is  developed  in 
the  concrete.  For  a  given  aggregate  and  a  given  amount  of  cement, 
a  decrease  in  the  amount  of  mixing  water  below  that  necessary  to  pro- 
duce sufficient  paste  to  occupy  most  of  the  voids  and  provide  the  lubri- 
cating layer  will  give  a  mix  deficient  in  mobility  and  lower  in  strength. 

"A  certain  degree  of  mobility  is  necessary  in  order  to  place  concrete 
in  the  forms  in  a  compact  and  solid  mass,  the  degree  varying  considerably 
with  the  nature  of  the  work  and  generally  it  will  be  found  necessary 
to  sacrifice  strength  to  secure  the  requisite  mobility.  It  is  readily  seen, 
however,  that  the  effort  should  be  made  to  produce  as  strong  a  cementing 
layer  of  paste  as  practicable  by  selecting  the  proper  mixture  of  ag- 
gregate and  by  regulating  the  amount  of  mixing  water. 

"More  thorough  mixing  not  only  mixes  the  paste  and  better  coats 
the  particles,  but  it  makes  the  mass  mobile  with  a  smaller  percentage 
of  mixing  water  and  this  less  dilute  paste  results  in  higher  strength. 
Any  improvement  in  methods  of  mixing  which  increases  the  mobility 
of  the  mass  will  permit  the  use  of  less  dilute  paste  and  thereby  secure 
increased  strength." 

In  connection  with  the  above  remarks  by  the  Dean  of  Concrete 
Investigators,  there  may  be  quoted  the  conclusions  of  a  classic  re- 
port prepared  by  the  Bureau  of  Standards.1 

"1.  No  standard  of  compressive  strength  can  be  assumed  or  guaran- 
teed for  concrete  of  any  particular  proportions  made  with  any  aggregate 
unless  all  the  factors  entering  into  its  fabrication  are  controlled. 

"  2.  A  concrete  having  a  desired  compressive  strength  is  not  neces- 
sarily guaranteed  by  a  specification  requiring  only  the  use  of  certain 
types  of  materials  in  stated  proportions.  Only  a  fractional  part  of 
the  desired  strength  may  be  obtained  unless  other  factors  are  controlled. 

"3.  The  compressive  strength  of  concrete  is  just  as  much  dependent 
upon  other  factors,  such  as  careful  workmanship  and  the  use  of  the 
proper  amount  of  water  in  mixing  the  concrete  as  it  is  upon  the  use 
of  the  proper  quantity  of  cement. 

"4.  The  compressive  strength  of  concrete  may  be  reduced  by  the 
use  of  an  excess  of  water  in  mixing  to  a  fractional  part  of  what  it  should 
attain  with  the  same  materials.  Too  much  emphasis  cannot  be  placed 
upon  the  injurious  effect  oj  the  use  oj  excessive  quantities  of  water  in  mixing 
concrete.  [The  italics  are  mine.] 

1  Technology  Papers  of  the  Bureau  of  Standards,  No.  58. 


200  RETAINING  WALLS 

"5.  The  compress! ve  strength  of  concrete  may  be  greatly  reduced 
if,  after  fabrication,  it  is  exposed  to  the  sun  and  wind  or  in  any  relatively 
dry  atmosphere  in  which  it  loses  its  moisture  rapidly,  even  though 
suitable  materials  were  used  and  proper  methods  of  fabrication  employed. 

"6.  The  relative  compressive  strengths  of  concretes  to  be  obtained 
from  any  given  materials  can  be  determined  only  by  an  actual  test 
of  those  materials  combined  in  a  concrete. 

"7.  Contrary  to  general  practice  and  opinion  the  relative  value  of 
several  fine  aggregates  to  be  used  in  concrete  can  not  be  determined  by 
testing  them  in  mortar  mixtures.  They  must  be  tested  in  the  combined 
state  with  the  coarse  aggregate. 

"8.  Contrary  to  general  practice  and  opinion  the  relative  value 
of  several  coarse  aggregates  to  be  used  in  concrete  cannot  be  determined 
by  testing  them  with  a  given  sand  in  one  arbitrarily  selected  proportion. 
They  should  be  tested  in  such  combination  with  the  fine  aggregate  as 
will  give  maximum  density,  assuming  the  same  ratio  of  cement  to 
total  combined  aggregate  in  all  cases. 

"9.  No  type  of  aggregate  such  as  granite,  gravel  or  limestone  can 
be  said  to  be  generally  superior  to  all  other  types.  There  are  good 
and  poor  aggregates  of  each  type. 

"10.  By  proper  attention  to  methods  of  fabricating  and  curing, 
aggregates  which  appear  inferior  and  may  be  available  at  the  site  of 
the  work  may  give  as  high  compressive  strength  in  concrete  as  the 
best  selected  materials  brought  from  a  distance,  when  the  latter  are 
carelessly  or  improperly  used. 

"11.  Density  is  a  good  measure  of  the  relative  compressive  strength 
of  several  different  mixtures  of  the  same  aggregates  with  the  same 
proportion  of  cement  to  the  total  aggregate.  The  mixture  having  the 
highest  density  need  not  necessarily  have  the  maximum  strength  but 
it  will  have  a  relatively  high  strength. 

"12.  Two  concretes  having  the  same  density  but  composed  of  dif- 
ferent aggregates  may  have  widely  different  compressive  strength. 

"13.  There  is  no  definite  relation  between  the  gradation  of  the  ag- 
gregates and  the  compressive  strength  of  the  concrete  which  is  applic- 
able to  any  considerable  number  of  different  aggregates. 

"14.  The  gradation  curve  for  maximum  compressive  strength, 
which  is  usually  the  same  as  for  maximum  density,  differs  for  each 
aggregate. 

"15.  With  the  relative  volumes  of  fine  and  coarse  aggregate  fixed, 
the  compressive  strength  of  a  concrete  increases  directly,  but  not  in  a 
proportionate  ratio  as  the  cement  content.  An  increase  in  the  ratio 
of  cement  to  total  fine  and  coarse  aggregates  when  the  relative  propor- 
tions of  the  latter  are  not  fixed  does  not  necessarily  result  in  an  increase 
in  strength,  but  may  give  even  a  lower  strength. 


CONCRETE  CONSTRUCTION  201 

"16.  The  compressive  strength  of  concrete  composed  of  given 
materials,  combined  in  definite  proportions  and  fabricated  and  exposed 
under  given  conditions  can  be  determined  only  by  testing  the  concrete 
actually  prepared  and  treated  in  the  prescribed  manner. 

"17.  The  results  included  in  this  paper  would  indicate  that  the  com- 
pressive strength  of  most  concretes,  as  commercially  made  can  be  increased 
25  to  100  per  cent,  or  more  by  employing  rigid  inspection  which  will  insure 
proper  methods  of  fabrication  of  the  materials." 

In  a  striking  report  on  how  to  properly  design  a  concrete 
mixture  to  obtain  the  utmost  strength  from  the  aggregate  at  hand 
by  Prof.  Duff  A.  Abrams1  it  is  shown  how  little  the  present  day 
standard  methods  of  proportioning  concrete  make  for  concrete 
strength.  The  importance  of  the  report  and  its  vital  conclusions 
justify  the  rather  lengthy  excerpts  below. 

The  general  problem  of  concrete  mixtures  has  been  defined 
in  the  report  as  follows  and  some  of  the  principles  following  a 
series  of  50,000  tests  are  noted  therein. 

"The  design  of  concrete  mixtures  is  a  subject  of  vital  interest  to  all 
engineers  and  constructors  who  have  to  do  with  concrete  work.  The 
problem  involved  may  be  one  of  the  following: 

"1.  What  mix  is  necessary  to  produce  concrete  of  proper  strength 
for  a  given  work? 

"2.  With  given  materials  what  proportion  will  give  the  best  con- 
crete at  minimum  cost? 

"3.  With  different  lots  of  materials  of  different  characteristics  which 
is  best  suited  for  the  purpose? 

"4.  What  is  the  effect  on  strength  of  concrete  from  changes  in  mix, 
consistency  or  size  and  grading  of  aggregate? 

"Proportioning  concrete  frequently  involves  selection  of  materials 
as  well  as  their  combination.  In  general,  the  question  of  relative  costs 
is  also  present." 

Of  the  different  methods  of  proportioning  concrete,  Prof. 
Abrams  has  noted  the  following  as  among  the  most  important: 

"1.  Arbitrary  selection,  such  as  1  :2  :4  mix,  without  reference  to  the 
size  or  grading  of  the  fine  and  coarse  aggregate; 

"2.  Density  of  aggregates  in  which  the  endeavor  is  made  to  secure 
an  aggregate  of  maximum  density; 

"3.  Density  of  concrete  in  which  the  attempt  is  made  to  secure 
concrete  of  maximum  density; 

1  Design  of  Concrete  Mixtures,  Bulletin  1,  Structural  Materials  Research 
Laboratory,  Lewis  Institute,  Chicago. 


202  RETAINING  WALLS 

"4.  Sieve  analysis,  in  which  the  grading  of  the  aggregates  is  made 
to  approximate  some  predetermined  sieve  analysis  curve  which  is 
considered  to  give  the  best  results;" 

"5.  Surface  area1  of  aggregates. 

"It  is  a  matter  of  common  experience  that  the  method  of  arbitrary  selec- 
tion in  which  fixed  quantities  of  fine  and  coarse  aggregates  are  mixed 
without  regard  to  the  size  or  grading  of  the  individual  materials,  is  far 
from  satisfactory.  Our  experiments  have  shown  that  the  other  methods 
mentioned  above  are  also  subject  to  serious  limitations.  We  have 
found  that  the  maximum  strength  of  concrete  does  not  depend  on  either 
an  aggregate  of  maximum  density  or  a  concrete  of  maximum  density, 
and  that  the  methods  that  have  been  suggested  for  proportioning  con- 
crete by  sieve  analysis  of  aggregates  are  based  on  an  erroneous  theory. 
All  of  the  methods  of  proportioning  concrete  which  have  been  proposed 
in  the  past  have  failed  to  give  proper  attention  to  the  water  content 
of  the  mix.  Our  experimental  work  has  emphasized  the  importance  of 
the  water  in  concrete  mixtures,  and  shown  that  the  water  is,  in  fact,  the 
most  important  ingredient,  since  very  small  variations  in  water  content 
produce  more  important  variations  in  the  strength  and  other  properties 
of  concrete  than  similar  changes  in  the  other  ingredients. 

After  performing  a  series  of  over  50,000  tests,  covering  a 
period  of  three  years,  Prof.  Abrams  has  established  the  following 
important  principles  in  regard  to  the  correct  design  of  a  concrete 
mix. 

"1.  With  given  concrete  materials  and  conditions  of  test  the  quantity 
of  mixing  water  determines  the  strength  of  the  concrete,  so  long  as  the 
mix  is  of  workable  plasticity.  / 

"2.  The  sieve  analysis  furnishes  the  only  Correct  basis  for  proportion- 
ing aggregates  in  concrete  mixtures. 

"3.  A  simple  method  of  measuring  the  effective  size  and  grading  of  an 
aggregate  has  been  developed.  This  gives  rise  to  a  function  known  as 
the  "fineness  modulus"2  of  the  aggregate. 

"4.  The  fineness  modulus  of  an  aggregate  furnishes  a  rational  method 
for  combining  materials  of  different  size  for  concrete  mixtures. 

"5.  The  sieve  analysis  curve  of  the  aggregate  may  be  widely  dif- 
ferent in  form  without  exerting  any  influence  on  concrete  strength. 

"6.  Aggregate  of  equivalent  concrete-making  qualities  may  be 
produced  by  an  infinite  number  of  different  gradings  of  a  given  material. 

"7.  Aggregates  of  equivalent  concrete-making  qualities  may  be 
produced  from  materials  of  widely  different  size  and  grading. 

1  See  end  of  chapter  for  a  definition  of  Surface  Area. 

2  See  end  of  chapter  for  a  complete  definition  of  the  fineness  modulus. 


CONCRETE  CONSTRUCTION  203 

"8.  In  general,  fine  and  coarse  aggregates  of  widely  different  size 
or  grading  can  be  combined  in  such  a  manner  as  to  produce  similar 
results  in  concrete. 

"9.  The  aggregate  grading  which  produces  the  strongest  concrete 
is  not  that  giving  the  maximum  density  (lowest  voids).  A  grading 
coarser  than  that  giving  maximum  density  is  necessary  for  highest 
concrete  strength. 

"10.  The  richer  the  mix,  the  coarser  the  grading  should  be  for  an 
aggregate  of  given  maximum  size;  hence,  the  greater  the  discrepancy 
between  maximum  density  and  best  grading. 

"11.  A  complete  analysis  has  been  made  of  the  water  requirements 
of  concrete  mixes.  The  quantity  of  water  required  is  governed  by  the 
following  factors : 

"(a)  The  condition  of  " workability "  of  concrete  which  must  be 
used — the  relative  plasticity  or  consistency; 

"  (b)  The  normal  consistency  of  the  cement; 

"  (c)  The  size  and  grading  of  the  aggregate — measured  by  the  fineness 
modulus; 

"  (d)  The  relative  volumes  of  cement  and  aggregate — the  mix; 

"  (e)  The  absorption  of  the  concrete; 

"  (/)  The  contained  water  in  aggregate. 

"12.  There  is  an  intimate  relation  between  the  grading  of  the  ag- 
gregate and  the  quantity  of  water  required  to  produce  a  workable 
concrete. 

"  13.  The  water  content  of  a  concrete  mix  is  best  considered  in  terms 
of  the  cement — water-ratio. 

"14.  The  shape  of  the  particle  and  the  quality  of  the  aggregate 
have  less  influence  on  the  .concrete  strength  than  has  been  reported  by 
other  experimenters." 

Prof.  Abrams  has  experimentally  determined  the  relation  be- 
tween the  water  content  and  the  strength  of  the  concrete  and 
reports  the  following  most  important  conclusions  together  with 
an  empiric  relation  between  the  two. 

"It  is  seen  at  once  that  the  size  and  grading  of  the  aggregate  and  the 
quantity  of  cement  are  no  longer  of  any  importance  except  in  so  far 
as  these  factors  influence  the  quantity  of  water  required  to  produce  a 
workable  mix.  This  gives  us  an  entirely  new  conception  of  the  function 
of  the  constituent  materials  entering  into  a  concrete  mix  and  is  the 
most  basic  principle  which  has  been  brought  out  in  our  studies  of 
concrete. 

"The  equation  of  the  curve  is  of  the  form 

A 


204 


RETAINING  WALLS 


where  S  is  the  compressive  strength  of  the  concrete  and  x  is  the  ratio 
of  the  volume  of  water  to  the  volume  of  cement  in  the  batch.  A  and 
B  are  constants  whose  values  depend  on  the  quality  of  the  cement  used, 
the  age  of  the  concrete,  curing  conditions,  etc. 

"This  equation  expresses  the  law  of  the  strength  of  concrete  so 
far  as  the  proportions  of  materials  are  concerned.  It  is  seen  that  for 
given  concrete  materials  the  strength  depends  upon  only  one  factor — 
the  ratio  of  water  to  cement.  Equations  which  have  been  proposed 
in  the  past  for  this  purpose  contain  terms  which  take  into  account  such 
factors  as  quantity  of  cement,  proportions  of  fine  and  coarse  aggregate, 
voids  in  aggregate,  etc.,  but  they  have  uniformly  omitted  the  only 
term  which  is  of  any  importance;  that  is,  the  water. 

"A  vital  function  entering  into  the  analysis  is  the  so-called  'fineness 
modulus'  which  may  be  defined  as  follows: 

"The  sum  of  the  percentages  in  the  sieve  analysis  of  the  aggregate 
divided  by  100. 

"The  sieve  analysis  is  determined  by  using  the  following  sieve  from 
the  Tyler  standard  series:  100,  48,  28,  14,  8,  4%,  %  and  1^  in.  These 
sieves  are  made  of  square-mesh  wire  cloth.  Each  sieve  has  a  clear 

TAQLE  36. — METHOD  OF  CALCULATING  FINENESS  MODULUS  OF  AGGREGATES 

The  sieves  used  are  commonly  known  as  the  Tyler  standard  sieves.  Each 
sieve  has  a  clear  opening  just  double  that  of  the  preceding  one. 

The  sieve  analysis  may  be  expressed  in  terms  of  volume  or  weight. 

The  fineness  modulus  of  an  aggregate  is  the  sum  of  the  precentages  given 
by  the  sieve  analysis,  divided  by  100. 


Size  of 

Sieve  analysis  of  aggregates 
per  cent,  of  sample  coarser  than  a  given  sieve 

Sieve 

Sand 

Pebbles 

>-1                     j 

square  opening 

Fine 

Medium 

Coarse 

Fine 

Medium 

Coarse 

aggregate 
(Gf)  * 

in. 

mm. 

(O 

(£) 

3) 

(F) 

100-mesh.... 

.0058 

.147 

82 

91 

97 

100 

100 

100 

98 

48-mesh.... 

.0116 

.295 

52 

70 

81 

100 

100 

100 

92 

28-mesh  

.0232 

.59 

20 

46 

63 

100 

100 

100 

86 

14-mesh  

.046 

1.17 

0 

24 

44 

100 

100 

100 

81 

8-mesh  

.093 

2.36 

0 

10 

25 

100 

100 

100 

78 

4-mesh.  .  .  . 

.185 

4.70 

0 

0 

0 

86 

95 

100 

71 

s^-in  

.37 

9.4 

0 

0 

0 

51 

66 

86 

49 

%-in  

.75 

18.8 

0 

0 

0 

9 

25 

50 

19 

JM-in  

1.5 

38.1 

0 

0 

0 

0 

0 

0 

0 

Fineness  modulus  .  . 

1.54 

2.41 

3.10 

6.46 

6.86 

7.36 

5.74 

*  Concrete  aggregate  G  is  made  up  of  25  per  cent,  of  sand  B  mixed  with  75  per  cent,  of 
pebbles  E.  Equivalent  gradings  would  be  secured  by  mixing  33  per  cent,  sand  B  with  67 
per  cent.  co»rse  pebbles  F\  28  'per  cent.  A  with  72  per  cent.  F,  etc.  The  proportion  coarser 
than  a  given  sieve  is  made  up  by  the  addition  of  these  percentages  of  the  corresponding  size 
of  the  constituent  materials. 


CONCRETE  CONSTRUCTION  205 

opening  just  double  the  width  of  the  preceding  one.  The  exact  di- 
mensions of  the  sieves  and  the  method  of  determining  the  fineness  mod- 
ulus will  be  found  in  Table  36.  It  will  be  noted  that  the  sieve  analysis 
is  expressed  in  terms  of  the  percentages  of  material  by  volume  or  weight 
coarser  than  each  sieve." 

Prof.  Abrams  notes  that  there  is  a  direct  relation  between 
the  fineness  modulus  as  above  defined  and  the  compressive 
strength  of  the  concrete,  after  noting  that  the  "  fineness  modulus 
simply  reflects  the  changes  in  water-ratio  necessary  to  produce  a 
given  plastic  condition."  This  is,  of  course,  consistent  with  his 
main  thesis  that  the  water-ratio  is  the  all  important  function  in 
determining  the  concrete  strength.  It  is  stated  that  the  relation 
between  the  compressive  strength  of  the  concrete,  as  brought 
out  by  tests  and  the  fineness  modulus  is  to  all  intents  a  linear  one, 
i.e.  an  increase  in  the  fineness  modulus  has  a  proportionate 
increase  in  the  compressive  strength. 

With  an  assigned  compressive  strength  of  concrete,  it  is  now 
possible  to  proceed  directly  to  assemble  an  aggregate  to  meet 
this  strength.  The  water-ratio  forming  the  fundamental  basis 
of  the  process,  the  empiric  relation  above  mentioned  is  employed 
to  determine  the  proper  value  of  x,  when  S  is  given  and  A  and 
B  are  known.  The  details  following,  showing  the  method  of 
obtaining  the  values  of  the  constants,  of  the  fineness  modulus 
and  of  the  several  combinations  possible  to  satisfy  most  economic- 
ally the  strength  requirements  of  the  concrete  are  given  with 
elegance  and  clearness  in  the  Bulletin  just  quoted.  The  novelty 
of  the  method  and  its  apparent  intricacy  (and  such  intricacy  is 
only  apparent)  and  the  fact  that  concrete  mixes  usually  just 
"grow"  and  are  not  scientifically  developed  may  make  Prof. 
Abrams'  procedure  seem  very  cumbersome.  A  little  study  of 
his  methods  will  show  that  the  contrary  is  true  and  that  the 
correct  design  of  a  concrete  mix  predicated  upon  his  assump- 
tions (and  these  assumptions  are  assuredly  based  on  most  valid 
premises)  is  a  matter  of  very  simple  analysis. 

The  further  comments  on  the  design  of  a  concrete  mix,  given 
at  the  conclusion  of  the  Bulletin  are  worthy  of  quotation  here: 

"The  importance  of  the  water-ratio  on  the  strength  of  concrete  will 
be  shown  in  the  following  considerations : 

"One  pint  more  water  than  necessary  to  produce  a  plastic  concrete 
reduces  the  strength  to  the  same  extent  as  if  we  should  omit  2  to  3 
Ib.  of  cement  from  a  one-bag  batch. 


206  RETAINING  WALLS 

"Our  studies  give  us  an  entirely  new  conception  of  the  function 
performed  by  the  various  constituent  materials.  The  use  of  a  coarse 
well-graded  aggregate  results  in  no  gain  in  strength  unless  we  take 
advantage  of  the  fact  that  the  amount  of  water  necessary  to  produce  a 
plastic  mix  can  thus  be  reduced.  In  a  similar  way  we  may  say  that 
the  use  of  more  cement  in  a  batch  does  not  produce  any  beneficial  effect 
except  from  the  fact  that  a  plastic  workable  mix  can  be  produced  with 
a  lower  water-ratio. 

"The  reason  a  rich  mixture  gives  a  higher  strength  than  a  lean  one 
is  not  that  more  cement  is  used,  but  because  the  concrete  can  be  mixed 
(and  usually  is  mixed)  with  a  water-ratio  which  is  relatively  lower 
for  the  richer  mixtures  than  for  the  lean  ones.  If  advantage  is  not  taken 
of  the  fact  that  in  a  rich  mix  relatively  less  water  can  be  used,  no  benefit 
will  be  gained  as  compared  with  a  leaned  mix.  In  all  this  discussion 
the  quantity  of  water  is  compared  with  the  quantity  of  cement  in  the 
batch  (cubic  feet  of  water  to  one  sack  of  cement)  and  not  to  the  weight 
of  dry  materials  or  of  the  concrete  as  is  generally  done. 

"The  mere  use  of  richer  mixes  has  encouraged  a  feeling  of  security, 
whereas  in  many  instances  nothing  more  has  been  accomplished  than 
wasting  a  large  quantity  of  cement,  due  to  the  use  of  an  excess  of  mixing 
water.  The  universal  acceptance  of  this  false  theory  has  exerted  a  most 
pernicious  influence  on  the  proper  use  of  concrete  materials  and  has 
proven  to  be  an  almost  insurmountable  barrier  in  the  way  of  progress 
in  the  development  of  sound  principles  of  concrete  proportioning  and 
construction. 

"Rich  mixes  and  well-graded  aggregates  are  just  as  essential  as  ever, 
but  we  now  have  a  proper  appreciation  of  the  true  function  of  the 
constituent  materials  in  concrete  and  a  more  thorough  understanding 
of  the  injurious  effect  of  too  much  water.  Rich  mixes  and  well-graded 
aggregates  are,  after  all,  only  a  means  to  an  end;  that  is,  to  produce  a 
plastic,  workable  concrete  with  a  minimum  quantity  of  water  as  com- 
pared with  the  cement  used.  Workability  of  concrete  mixes  is  of 
fundamental  significance.  This  factor  is  the  only  limitation  which 
prevents  the  reduction  of  cement  and  water  to  much  lower  limits  than 
are  now  practicable. 

"The  above  considerations  show  that  the  water  content  is  the  most 
important  element  of  a  concrete  mix,  in  that  small  variation  in  the 
water  cause  a  much  wider  change  in  the  strength  than  similar  variations 
in  the  cement  content  or  the  size  or  grading  of  the  aggregate.  This 
shows  the  absurdity  of  our  present  practice  in  specifying  definite  grad- 
ings  for  aggregates  and  carefully  proportioning  the  cement,  then  guessing 
at  the  water.  (The  italics  are  mine.)  It  would  be  more  correct  to 
carefully  measure  the  water  and  guess  at  the  cement  in  the  batch. 

"The  grading  of  the  aggregate  may  vary  over  a  wide  range  without 


CONCRETE  CONSTRUCTION  207 

producing  any  effect  on  concrete  strength  so  long  as  the  cement  and 
water  remain  unchanged.  The  consistency  of  the  concrete  will  be 
changed,  but  this  will  not  affect  the  concrete  strength  if  all  mixes  are 
plastic.  The  possibility  of  improving  the  strength  of  concrete  by 
better  grading  of  aggregates  is  small  as  compared  with  the  advantages 

which  may  be  reaped  from  using  as  dry  a  mix  as  can  be  properly  placed. 
********** 

"Without  regard  to  actual  quantity  of  mixing  water  the  following 
rule  is  a  safe  one  to  follow:  Use  the  smallest  quantity  of  mixing  water 
that  will  produce  a  plastic  or  workable  concrere.  The  important  of  any 
method  of  mixing,  handling,  placing  and  finishing  concrete  which  will 
enable  the  builder  to  reduce  the  water  content  of  the  concrete  to  a 
minimum  is  at  once  apparent." 

Practical  Application. — Some  of  the  details  of  these  copious 
excerpts  may  eventually  prove  without  adequate  experimental 
basis;  yet  the  fundamental  truth  conveyed  in  all  the  foregoing 
must  be  recognized — namely,  the  role  of  the  water  content  of  a 
concrete  mix.  The  question  of  paramount  importance  is  the 
manner  and  means  of  applying  these  truths  to  actual  concrete 
work  in  the  field.  Stone,  gravel,  sand  and  cement  companies 
have  been  educated  to  furnish  products  meeting  with  the  require- 
ments of  long  continued  experimental  and  field  research.  These 
products  are  naturally  much  costlier  than  are  aggregates  unre- 
stricted as  to  nature,  impurities,  grading  and  size.  It  is  essential 
then  that  this  added  cost  be  not  squandered  without  any  benefit 
through  oversight  of  some  simple  principles. 

The  proper  mixing  of  the  ingredients  is  conditioned  upon  the 
plant  used,  both  for  mixing  and  for  distributing.  The  character 
of  such  plant  has  been  described  both  generally  and  in  some  detail 
in  a  previous  chapter  on  plant.  The  average  mixer,  while  a  more 
or  less  efficient  machine  has  some  difficulty  in  producing  a  well 
mixed  batch  of  low  water  content  in  a  short-timed  mix.  A  little 
patience  in  educating  the  mixer  operator  to  keep  the  water  con- 
tents low  and  an  insistence  that  the  concrete  be  not  dumped 
until  a  specified  time  of  mixing  has  elapsed,  will  go  a  long  way 
towards  meeting  the  experimental  requirements  of  good  concrete. 
Clearly,  it  is  of  no  avail  to  go  to  the  bother,  expense  and  the  pos- 
sible delay  of  securing  specified  concrete  materials,  if  little  atten- 
tion is  paid  to  the  final  steps  in  concrete  mixing. 

A  batch  of  concrete  must  be  in  the  mixer  a  certain  minimum 
time  before  the  aggregate  has  been  properly  transformed  into 


208  RETAINING  WALLS 

concrete.  What  this  time  is  depends  upon  the  character  of  the 
machine  and  the  number  of  revolutions  it  makes  per  minute. 
This  time  can  not  be  specified  in  advance  nor  can  good  concrete 
be  expected  merely  from  long  time  mixing.  In  this  connection 
see  the  Engineering  News-Record,  Nov.  28,  1918,  p.  966,  and 
Jan.  23,  1919,  p.  200.  The  average  time  of  mixing  a  batch  is 
about  one  minute.  A  little  care  and  study  of  the  particular 
machine  at  hand  will  determine  the  correct  time  for  a  batch  mix. 
Careful  inspection  will  then  insure  that  each  batch  of  concrete 
will  receive  this  length  of  time  for  its  proper  mix. 

In  the  use  of  small  mixers,  the  so-called  one  or  two  bag  batch 
mixers,  it  is  exceedingly  hard  to  get  a  uniform  water  ratio  for  all 
the  batches.  Variations  in  the  piling  of  the  stone  and  sand  in 
the  barrows ;  in  the  dryness  of  the  aggregate  all  make  it  impossible 
to  apply  a  constant  amount  of  water  and  turn  out  the  same  con- 
sistency of  mix.  However,  by  a  careful  attention  to  the  piling 
of  the  carts  and  by  an  insistence  that  water  be  used  in  measured 
quantity  only — preferably  from  an  overhead  tank  attached  to 
the  machine  and  certainly  not  by  an  indiscriminate  use  of  the 
hose  or  pail — a  concrete  can  be  obtained  meeting  with  a  fair 
degree  of  success  the  water"  requirements  of  workable  plastic 
concrete. 

It  should  be  definitely  predicated  that  the  principles  of  good 
concrete  should  determine  the  plant  and  not,  conversely,  the 
plant  determine  the  mode  of  concreting  (see  chapter  on  Plant). 

Concrete  Methods. — The  question  of  competent  labor  proves 
a  most  irritating  one.  It  may  be  set  down  as  axiomatic  that 
common  labor,  however  willing,  and  in  spite  of  competent  leader- 
ship cannot  mix  and  place  good  concrete.  A  trained  concrete 
force  is  necessary  for  this  work.  The  use  of  incompetent  labor 
on  concrete  work  is  a  most  short-sighted  policy  and  here,  as  in 
every  other  industrial  enterprise,  the  best  is  decidedly  the  cheap- 
est in  the  end. 

The  use  of  poor  materials  and  the  employment  of  lax  and  in- 
different methods  together  with  incompetent  labor  are  dependent 
upon  the  laxity  of  inspection  and,  unfortunately,  the  minimum 
requirements  of  the  engineer  form  the  maximum  goal  of  the  aver- 
age contractor  and,  to  use  the  colloquialism  of  the  field,  the  con- 
struction superintendent  will  "get  away  with"  as  much  as  he 
can.  True,  there  are  many  exceptions,  but  the  engineer  does 
well  to  prepare  for  the  worst. 


CONCRETE  CONSTRUCTION  209 

To  specify  a  good  concrete,  especially  in  light  of  the  above 
researches,  is,  comparatively  an  easy  matter.  To  assign  proper 
inspection,  tempered  by  practical  judgment  and  equipped  with 
a  thorough  knowledge  of  good  concrete,  so  that  in  matters  of 
field  decision  the  concrete  is  given  the  benefit  of  the  doubt,  is  a 
far  more  difficult  matter. 

As  the  details  of  the  requirements  of  good  concrete  become  more 
generally  known  undoubtedly  the  common  welfare  of  the  con- 
crete interests,  contractors,  engineers,  plant  manufacturers  and 
the  like,  will  promote  a  cooperation  that  will  make  it  a  much  simpler 
matter  to  secure  the  maximum  strength  of  concrete  from  a  given 
assembly  of  materials.  At  present  it  is  necessary  to  specify  in 
detail  the  desired  concrete  aggregates  and  the  methods  by  which 
these  are  to  be  mixed  and,  in  addition,  to  make  ample  provision 
for  carrying  out  the  intent  and  letter  of  the  specifications. 

Distributing  Concrete. — Concrete,  properly  mixed,  must  like- 
wise be  properly  distributed.  Poor  distribution  will  nullify  the 
beneficial  results  of  good  mixing.  The  concrete  mix  is  an  aggre- 
gate of  solids  in  a  fluid  vehicle  and,  when  transported  in  any  but 
a  vertical  direction,  will  tend  to  separate  in  accordance  with 
natural  laws.  The  distributing  system  must  aid  in  overcoming 
this  separation  tendency.  For  this  reason  concrete  should  be 
dropped  vertically  into  the  forms  and  spread  by  shovels  and  hoes 
into  thin  layers.  Spouting  a  concrete  into  a  form  in  any  direc- 
tion but  the  vertical  is  a  serious  offence.  The  mix  will  separate 
and  any  subsequent  hoeing,  shovelling  or  spading  will  prove  inef- 
fectual. Upon  stripping  the  forms  the  inevitable  pouring  streaks 
will  appear;  evidence  of  poor  workmanship  and  presenting  a 
most  unpleasing  appearance. 

With  a  concrete  of  workable  plasticity,  properly  delivered 
into  a  form,  but  little  additional  work  should  be  necessary  to 
bring  it  to  its  final  place  in  the  form.  The  concrete  should  be 
spaded  at  the  form  to  permit  the  grout  to  collect  at  the  face,  in- 
suring a  smooth  face  and  should  also  be  spaded  at  the  rods  to  aid 
in  getting  a  firm  grout  bond  between  the  steel  and  the  concrete. 

The  distributing  systems  have  been  discussed  in  detail  in  the 
preceding  chapter  on  plant,  which  chapter  should  be  read  again 
in  the  light  of  the  present  observations  upon  the  requirements  of 
good  concrete. 

Keying  Lifts. — If  the  day's  pour  is  finished  before  reaching 
the  top  of  the  wall,  the  concrete  surface  should  be  brought  to  a 

14 


210  RETAINING  WALLS 

rough  level  and  a  long  timber  to  form  a  longitudinal  key  should 
be  imbedded  in  the  top.  Dowels  may  be  inserted  instead,  made 
up  either  of  steel  rods,  or  of  stones  and  carried  about  one  foot 
into  each  of  the  layers.  At  the  pouring  of  the  next  layer,  the 
timber  key,  if  used,  is  to  be  removed,  the  surface  to  be  thoroughly 
cleaned  and  the  fresh  concrete  then  placed  upon  it.  For  the 
efficiency  of  various  treatments  of  this  joint  see  "  Construction 
Joints, "  page  159. 

Use  of  Cyclopean  Concrete. — In  large  concrete  walls,  it  is  per- 
missible to  place  stones  over  12  inches  in  diameter  wherever  the 
thickness  of  the  concrete  mass  exceeds  30  inches.  The  stones 
are  kept  about  12  inches  apart  and  about  6  inches  from  the  face 
of  the  wall.  They  should  be  sound,  hard  rock,  well-cleaned 
and  should  be  placed  by  hand  into  the  concrete  and  not  dumped 
indiscriminately  from  a  bucket  or  thrown  in  at  random.  A  little 
care  in  placing  the  stone  will  permit  a  larger  number  to  be  used 
and  thus  cut  down  the  cost  of  the  wall  by  economizing  on  the 
amount  of  concrete  aggregate  required. 

In  reinforced  concrete  walls  it  is  questionable  whether  the  use 
of  such  " plums"  should  be  permitted.  The  rod  system  makes 
it  difficult  to  place  the  stones,  even  though  the  wall  exceeds  30 
inches  in  thickness.  Since  the  concrete  in  this  wall  is  highly 
stressed  in  compression,  sound  rock  must  be  used.  With  a  care- 
fully specified  aggregate  for  the  concrete,  it  seems  a  little  incon- 
sistent then  to  permit  the  use  of  an  indeterminate  material. 
Local  conditions  will  generally  indicate  whether  good  stones  are 
available.  As  a  general  rule,  however,  for  the  usual  type  of 
cantilever  and  counterforted  walls,  the  use  of  plums  is  inadvisable. 

Winter  Concreting. — Quite  often  the  urgent  need  of  a  concrete 
retaining  wall  makes  it  imperative  that  its  construction  proceed 
despite  winter  weather.  As  the  temperature  drops,  the  setting 
time  of  concrete  increases.  The  setting  action  stops  when  the 
concrete  is  frozen  and  does  not  continue  until  the  concrete  has 
thawed.  It  is  doubtful  whether  frost  injures  a  concrete  perma- 
nently. This  much,  however,  is  certain — a  frozen  concrete 
must  thaw  out  completely  and  then  be  given  ample  time  to  set, 
before  the  forms  are  stripped  or  any  load  placed  upon  the  wall. 
It  is  highly  desirable  and  it  is  generally  so  specified  that  concrete 
be  mixed  in  such  a  manner  that  it  reaches  the  form  at  a  favorable 
setting  temperature  and  is  then  to  be  suitably  protected  against 
frost  until  it  is  thoroughly  set. 


CONCRETE  CONSTRUCTION  211 

Concrete  should  not  reach  the  forms  at  a  temperature  less  than 
45°  (Fahrenheit) .  The  aggregate  and  the  water  should  be  heated 
when  the  temperature  drops  below  this  mark.  While,  ordinarily, 
concreting  is  permitted  without  heating  the  materials  until  the 
temperature  drops  below  the  freezing  point,  the  above  tempera- 
ture should  preferably  be  the  controlling  one. 

A  simple  method  of  heating  the  aggregate  is  to  pile  it  around 
a  large  metal  pipe  (a  large  diameter  metal  flue,  or  a  water  pipe  is 
just  the  thing)  and  have  a  fire  going  within  the  pipe.  Old  form 
lumber  is  an  excellent  and  cheap  fuel  for  this  fire.  Another, 
similar  method  is  to  pile  the  material  on  large  metal  sheets  rest- 
ing on  little  stone  piers,  and  beneath  which  sheets  fires  are  kept 
burning.  In  both  the  methods  care  must  be  taken  not  to  burn 
the  material  next  to  the  metal,  and  not  to  use  such  material  if  it 
does  become  burned.  The  water  may  be  heated  in  large  con- 
tainers over  fires,  or  by  passing  live  steam  through  the  water, 
either  directly  in  it  or  through  coils. 

An  interesting  description  of  a  winter  concreting  job  is  given 
here  r1 

"The  sand  and  crushed  stone  used  in  making  the  wall  concrete  were 
heated  by  diffusion  of  steam  from  perforations  in  a  coil  of  a  2"  pipe 
placed  at  the  bottom  of  the  storage  pile.  The  bottoms  of  the  charging 
bin  above  the  mixer  were  also  fitted  with  perforated  piping  so  that  the 
heat  might  be  retained  in  the  materials. 

"The  water  used  in  mixing  was  maintained  at  about  100°  F.  by  a 
live  steam  jet  discharging  at  the  bottom  of  a  3000  gallon  tank,  or 
reservoir  kept  constantly  full.  The  overflow  from  the  tank  discharged 
into  a  50  gallon  measuring  barrel,  being  heated  to  scalding  temperature 
by  another  jet  of  superheated  steam. 

"The  walls  forms  were  insulated  with  straw  and  plank  on  the  back 
and  covered  with  tongue  and  grooved  flooring  on  the  face,  retaining  a  2" 
space  between  the  steel  (metal  forms  were  used)  and  the  wood,  through 
which  low  pressure  steam  from  one  of  the  boilers  on  the  deck  was  diffused 
by  a  perforated  I"  pipe.  This  pipe  was  at  the  bottom  of  the  form  and 
ran  longitudinally  the  entire  length  connecting  with  the  boiler  by  a  T 
connection  and  vertical  pipe  at  about  the  middle  of  the  section. 

"A  stationary  mixing  plant  was  installed  adjacent  to  the  main  line 
of  the  railway  about  half  a  mile  west  of  the  wall  site.  The  concrete 
was  conveyed  to  the  wall  in  buckets  on  cars  drawn  by  a  dinkey  on  narrow 
gage." 

1  Retaining  Walls,  Baltimore  &  Ohio  Railroad,  Engineering  News,  Vol.  76, 
p.  269. 


212  RETAINING  WALLS 

A  general  note  on  winter  concreting  on  Miami  Conservancy 
Work  is  given  here  as  of  interest  in  connection  with  the  present 
topic.1 

"Concreting  has  been  carried  on  through  the  winter  in  the  dam 
construction  work  of  the  Miami  Conservancy  District,  Ohio,  with  only 
occasional  interruption.  As  the  nature  of  the  enterprise  demands  that 
progress  be  rapid  and  according  to  schedule,  and  as  it  is  important  to 
keep  the  working  organization  intact  to  avoid  losses  and  delays,  it 
became  necessary  to  plan  reducing  the  interruptions  of  concreting  to  a 
minimum. 

"  Study  of  the  extra  costs  involved  in  heating  materials  and  protect- 
ing deposited  concrete  led  to  the  conclusion  that  the  greater  part  of  the 
extra  cost  is  incurred  only  at  temperatures  below  20°,  and  a  general  rule 
was  therefore  made  that  work  through  the  cold  season  is  to  be  continued 
until  the  thermometer  drops  below  20°. 

"Provision  for  heating  aggregates  by  steam  coils  built  in  the  bins  has 
been  made  at  all  three  of  the  dams  where  concreting  has  been  going  on 
*  *  *  .  Means  have  also  been  provided  for  protecting  the  surfaces  from 
freezing  by  tarpaulins  and  salamanders,  or,  in  some  instances  by  steam 
coils  (where  steam  was  available  because  it  was  used  for  other 
purposes). 

"  Care  is  taken  that  no  fresh  concrete  is  placed  on  frozen  foundations. 
With  a  view  to  reducing  the  liability  of  freezing  also,  the  amount  of 
water  used  in  the  mixing  is  closely  regulated." 

Concrete  work  in  winter,  observing  the  necessary  precautions 
to  prevent  freezing,  is,  of  course,  more  costly,  than  work  at  the 
seasonable  temperatures.  Whether  this  extra  cost  is  less  than 
the  loss  involved  in  the  break  in  the  continuity  of  the  work  and 
the  delay  in  receiving  the  finished  structure,  is  a  matter  to  be 
disposed  of  uniquely  for  each  piece  of  work.  If  the  work  is  to 
proceed  regardless  of  the  weather,  the  specifications  must  so 
be  drawn,  that  the  precautions  to  be  used  when  the  temperature 
falls  below  a  given  point  (which  must  be  clearly  noted)  are  em- 
phatically set  forth.  General  specifications  as  to  heating  are 
unsatisfactory — the  details  should  be  given. 

Acceleration  of  Concrete  Hardening. — The  quicker  a  concrete 
sets,  other  things  being  equal,  the  quicker  the  forms  can  be  strip- 
ped and  the  sooner  can  the  fill  be  deposited  behind  the  wall. 
Under  natural  conditions,  the  warmer  the  concrete  is  the  quicker 
it  sets.  Therefore  work  in  the  summer  can  proceed  at  a  faster 

1  Engineering  -News-Record,  Vol.  82,  p.  618. 


CONCRETE  CONSTRUCTION  213 

rate  than  work  at  the  other  seasons.  Some  cements  are  more 
quickly  setting  than  others.  It  is  possible,  by  adding  certain 
chemicals  to  accelerate  the  hardening  of  the  concrete.  The 
effect  of  the  addition  of  calcium  chloride  has  been  noted  as 
follows:1 

"As  the  result  of  some  experiments  made  by  the  Bureau  of  Standards 
to  develop  a  method  to  accelerate  the  rate  at  which  concrete  increases 
in  strength  with  age,  it  was  found  that  the  addition  of  small  quantities 
of  calcium  chloride  to  the  mixing  water  gave  the  most  effective  results. 
A  comprehensive  series  of  tests  was  inaugurated  to  determine  further 
the. amount  of  acceleration  in  the  strength  of  concrete  obtained  in  this 
manner  and  to  study  the  effect  of  such  additions  on  the  durability  of 
concrete  and  the  effect  of  the  addition  of  this  salt  on  the  liability  to  corro- 
sion of  iron  or  steel  imbedded  in  mortar  or  concrete. 

"The  results  to  date  indicate  that  in  concrete  at  the  age  of  two  or 
three  days,  the  addition  of  calcium  chloride  up  to  10  per  cent,  by  weight 
of  water  to  the  mixing  water  results  in  an  increase  in  strength,  over  simi- 
lar concrete  gaged  with  plain  water,  of  from  30  to  100  per  cent.,  the 
best  results  being  obtained  when  the  gaging  water  contains  from  4  to  6 
per  cent,  of  calcium  chloride. 

"Compressive  strength  tests  of  concretes  gaged  with  water  containing 
up  to  10  per  cent,  calcium  chloride,  at  the  age  of  one  year  gave  no  indi- 
cation that  the  addition  of  this  salt  had  a  deleterious  effect  on  the  dura- 
bility of  the  concrete. 

"  Corrosion  tests  that  have  been  completed  indicate  that  the  presence 
of  calcium  chloride,  although  the  amount  used  is  relatively  small,  in 
mortar  slabs  exposed  to  the  weather,  causes  appreciable  corrosion  of  the 
metal  within  a  year.  This  appears  to  indicate  that  calcium  chloride 
should  not  be  used  in  stuccos  and  warns  against  the  unrestricted  use  of 
this  salt  in  reinforced  concrete  exposed  to  weather  or  water." 

Concrete  Materials. — Concrete  aggregates  and  cement  have 
been  so  well  classified  and  placed  under  standard  specifications 
that  any  typical  specification  will  serve  as  a  model  for  the  charac- 
ter of  the  material  to  enter  into  the  construction  of  a  retain- 
ing wall.  A  brief  description  may  be  given  of  the  essential 
requirements  of  these  concrete  constituents.  It  may  be  well  to 
read  once  more  the  previous  pages  upon  the  bearing  of  the  type 
of  the  aggregate  on  the  concrete  strength  and  the  relative  im- 
portance of  the  character  and  proportions  of  the  aggregates 
(including  water)  as  compared  with  the  methods  of  preparation 

1  Engineering  News  Record,  Vol.  82,  p.  507. 


214 


RETAINING  WALLS 


and  distributing.  The  amounts  of  the  material  required  depend 
upon  the  proportions  specified.  Table  37  is  given  here  based 
upon  the  standard  proportion  and  shows  the  amount  of  cement, 
sand  and  stone  required  for  the  various  mixes.  These  are  the 
theoretical  requirements.  It  must  be  borne  in  mind  that  the 
method  of  distributing  the  material,  whether  in  central  bins  or 
in  local  piles  (see  chapter  preceding  on  " Plant")  will  involve  a 
certain  amount  of  wastage  which  must  be  taken  into  consider- 
ation in  ordering  the  aggregate.  Properly  constructed  shacks  for 
the  storage  of  cement  will  reduce  to  a  minimum  the  loss  of  ce- 
ment through  accidental  weathering,  etc. 

TABLE  37. — PROPORTIONS  FOR  MIXING  CONCRETE 


Mixtures 


Yardages  of  materials  for  one  cubic  yard  of  concrete 
in  the  form 


Specification  stone 
up  to  2  in. 


Gravel, 
±  in.  size 


cement 

oana 

otone 

Cement, 
bbls. 

Sand, 
yds. 

Stone, 
yds. 

Cement, 
bbls. 

Sand, 
yds. 

Stone, 
yds. 

1 

1.0 

2 

2.6      \       A 

.8. 

2.3 

.4 

.7 

1 

1.0 

3 

2.1 

.3    ;    .9 

1.9 

.3 

.9 

1 

1.5 

3 

1.9 

.4 

.8 

1.7 

.4 

.8 

1 

1.5 

4 

1.6 

.4 

1.0 

1.5 

.3 

.9 

1 

2.0 

3 

1.7 

.5 

.8 

1.5 

.5 

.7 

1 

2.0 

4 

1.5          .4 

.9 

1.3           .4 

.8 

1 

2.0 

5 

1.3           .4 

1.0 

1.2 

.4 

.9 

1 

2.5 

5 

1.2 

.5 

.9 

1.1 

.4 

.8 

1 

3.0 

4 

1.3 

.6 

.8 

1.2 

.5 

.7 

1 

3.0 

6 

1.0 

.5 

.9 

.9 

.4 

.8 

1 

3.5 

5 

1.1 

.6 

.8 

1.0 

.5 

.8 

1 

3.5 

7 

0.9 

.5 

.9 

.8 

.4 

.9 

1 

4.0 

6 

0.9 

.6             .8 

.8 

.5 

.8 

1 

4.0 

8 

0.8 

.5             .9 

.7 

.4 

.9 

1 

Cement. — (Portland  cement,  alone  is  discussed  here.)  It  is 
usual  to  specify  that  cement  will  meet  the  requirements  of  the 
Committee  of  the  American  Society  of  Civil  Engineers  on  "  Uni- 
form Tests  of  Cement. "  It  is  usual  to  insist  that  the  brand  of 
cement  used  is  one  that  has  been  employed  on  large  engineering 
works  for  at  least  five  years. 

Portland  cement  has  been  defined  as  the  finely  pulverized 
product  resulting  from  the  calcination  to  incipient  fusion  of  the 


CONCRETE  CONSTRUCTION  215 

properly  proportioned  mixture  of  argillaceous  and  calcareous 
materials  to  which  no  addition  greater  than  3  per  cent,  has  been 
made  subsequent  to  calcination. 

Its  fineness  shall  be  determined  and  limited  as  follows:  The 
cement  shall  leave  by  weight  a  residue  of  not  more  than  8  per 
cent,  on  a  No.  100  sieve  and  not  more  than  25  per  cent,  on  a  No. 
200  sieve,  the  wires  of  the  sieve  being  respectively  0.0045  and 
0.0024  of  an  inch  in  diameter. 

The  time  of  setting  shall  be  as  follows:  The  cement  shall 
develop  initial  set  in  not  less  than  30  minutes,  and  shall  develop 
hard  set  in  not  less  than  1  hour,  nor  more  than  10  hours. 

The  minimum  requirements  for  tensile  strength  for  briquettes 
one  inch  square  in  minimum  section  shall  be  as  follows : 

HEAT  CEMENT 

Age  Strength 

24  hours  in  moist  air 175  Ib. 

7  days  (1  day  in  moist  air,  6  days  in  water) 500  Ib. 

28  days  (1  day  in  moist  air,  27  days  in  water) 600  Ib. 

ONE  PART  CEMENT,  THREE  PARTS  STANDARD  SAND 

7  days  (1  day  in  moist  air,  6  days  in  water) 170  Ib. 

28  days  (1  day  in  moist  air,  27  days  in  water) 225  Ib. 

Neat  briquettes  shall  show  a  minimum  increase  in  strength 
of  10  per  cent,  and  sand  briquettes  20  per  cent,  from  the  tests  at 
the  end  of  7  days,  to  those  at  28  days. 

Tests  for  constancy  of  volume  will  be  made  by  means  of  pats 
of  neat  cement  about  3  inches  in  diameter,  J^  inch  thick  at  the 
center  and  tapering  to  a  thin  edge.  These  pats  to  satisfactorily 
answer  the  requirements  shall  remain  firm  and  hard  and  show 
no  signs  of  distortion,  checking,  cracking,  or  disintegrating. 

The  cement  shall  contain  not  more  than  1.75  per  cent,  of  anhy- 
drous sulphuric  acid  (S03),  or  more  than  4  per  cent,  of  magnesis 
(MgO). 

The  cement  shall  have  a  specific  gravity  of  not  less  than  3.10 
nor  more  than  3.25  after  being  thoroughly  dried  at  a  temperature 
of  212°F.  The  color  shall  be  uniform,  bluish  gray,  free  from  yel- 
low or  brown  particles. 

Sand. — Sand  for  concrete  shall  be  clean,  containing  not  more 
than  3  per  cent,  of  foreign  matter.  It  should  be  reasonable  free 
from  loam  and  dirt.  When  rubbed  between  the  palm  the  hand 
should  be  left  clean.  It  should  be  well  graded  from  coarse  to 
fine.  No  grains  should  be  left  on  a  ^-inch  sieve  and  not  more 


216  RETAINING  WALLS 

than  6  per  cent,  should  pass  through  a  100  mesh  sieve.  Fine 
sand  is  undesirable  and  its  presence  in  a  quantity  greater  than 
that  just  specified  will  materially  weaken  the  concrete.  A  coarse 
smooth-grained  sand  is  not  objectionable  and  will  produce,  with 
other  things  being  equal,  an  effective  and  strong  concrete.  In 
connection  with  the  selection  of  the  aggregate  and  the  proportion- 
ing of  the  coarse  and  fine  particles,  a  note  in  the  appendix  is 
given  on  the  selection  and  mixing  of  aggregates  by  the  surface 
area  method  and  by  the  fineness  modulus  method  and  the  rela- 
tion between  these  two  modes  of  selection  and  the  strength  of 
the  concrete.1 

Crushed  Stone  and  Gravel. — Crushed  stone  should  be  made 
from  trap  or  limestone.  Stone  from  local  quarries,  or  from 
rock  cuts  encountered  in  the  work  should  be  used  only  after 
tests  have  been  made  on  concrete  containing  this  stone.  For 
ordinary  gravity  walls,  the  size  of  the  crushed  stone  or  of  the 
gravel  may  vary  from  %  inch  to  1%  inch  in  diameter.  For  the 
thin  reinforced  concrete  walls  the  stone  should  not  exceed  % 
inch  in  size. 

Occasionally  the  sand  and  the  stone  are  delivered  already 
mixed  in  the  required  proportions.  Parallel  to  this  method, 
the  run  of  a  gravel  bank  may  be  taken,  including  the  gravel 
with  the  finer  sands.  Either  method  of  supplying  the  aggregate 
is  far  from  ideal  and  does  not  lend  itself  well  to  a  conscientious 
proportioning  of  the  materials.  It  is  preferable  to  supply  the 
coarse  and  the  fine  aggregates  separately  and  mix  them  in  the 
required  proportions  in  the  mixer. 

A  resume  of  the  above  methods  of  selecting  the  aggregates 
and  cement  is  presented  in  the  appendix  in  the  shape  of  a  standard 
specification  for  retaining  walls,  including  the  proper  specifying 
of  the  materials  entering  into  its  composition. 

Fineness  Modulus  of  Aggregate.2 — The  experimental  work  car- 
ried out  in  the  laboratory  has  given  rise  to  what  we  term  the 
fineness  modulus  of  the  aggregate.  It  may  be  defined  as  fol- 
lows: The  sum  of  the  percentages  in  the  sieve  analysis  of  the 
aggregate  divided  by  100. 

The  sieve  analysis  is  determined  by  using  the  following  sieves 

1  See  preceding  pages  on  the  fineness  modulus;  also  Engineering  News- 
Record,  June  12,  1919,  pp.  1142  to  1149. 

2  Bulletin    No.    1,    Structural    Materials    Research    Laboratory,   Lewis 
Institute,  Chicago,  D.  A.  Abrams. 


CONCRETE  CONSTRUCTION 


217 


from  the  Tyler  standard  series:  100,  48,  28,  14,  8,  4,  %-in., 
%-in.  and  l^-in.  These  sieves  are  made  of  square-mesh  wire 
cloth.  Each  sieve  has  a  clear  opening  just  double  the  width 
of  the  preceding  one.  The  exact  dimensions  of  the  sieves  and 
the  method  of  determining  the  fineness  modulus  will  be  found  in 
Table  36.  It  will  be  noted  that  the  sieve  analysis  is  expressed  in 
terms  of  the  percentages  of  material  by  volume  or  weight  coarser 
than  each  sieve. 

A  well-graded  torpedo  sand  up  to  No.  4  sieve  will  give  a  fineness 
modulus  of  about  3 .00 ;  a  coarse  aggregate  graded  4-lJ^-m.  will  give 
fineness  modulus  of  about  7.00;  a  mixture  of  the  above  materials 
in  proper  proportions  for  a  1 :4  mix  will  have  a  fineness  modulus  of 
about  5.80.  A  fine  sand  such  as  drift-sand  may  have  a  fineness 
modulus  as  low  as  1.50. 


100 


43 


28  14  6  4 

Sieve  Si_ze(Lccj. Scale} 


FIG.  120.— From  Bulletin  No.  1.     D.  A.  Abrams,  Structural  Materials  Research 
Laboratory,  Lewis  Institute,  Chicago. 


Sieve  Analysis  of  Aggregates. — There  is  an  intimate  relation 
between  the  sieve  analysis  curve  for  the  aggregate  and  the  fineness 
modulus;  in  fact,  the  fineness  modulus  enables  us  for  the  first 
time  to  properly  interpret  the  sieve  analysis  of  an  aggregate. 


218 


RETAINING  WALLS 


If  the  sieve  analysis  of  an  aggregate  is  platted  in  the  manner 
indicated  in  Fig.  120  that  is,  using  the  per  cent,  coarser  than  a 
given  sieve  as  ordinate,  and  the  sieve  size  (platted  to  logarithmic 
scale)  as  abscissa,  the  fineness  modulus  of  the  aggregate  is  mea- 
sured by  the  area  below  the  sieve  analysis  curve  The  dotted 
rectangles  for  aggregate  "G"  show  how  this  result  is  secured. 
Each  elemental  rectangle  is  the  fineness  modulus  of  the  material 
of  that  particular  size.  The  fineness  modulus  of  the  graded 
aggregate  is  then  the  summation  of  these  elemental  areas.  Any 
other  sieve  analysis  curve  which  will  give  the  same  total  area 
corresponds  to  the  same  fineness  modulus  and  will  require  the 
same  quantity  of  water  to  produce  a  mix  of  the  same  plasticity 
and  gives  concrete  of  the  same  strength,  so  long  as  it  is  not  too 
coarse  for  the  quantity  of  cement  used. 

The  fineness  modulus  may  be  considered  as  an  abstract  num- 
ber; it  is  in  fact  a  summation  of  volumes  of  material.  There  are 
several  different  methods  of  computing  it,  all  of  which  will  give 
the  same  result.  The  method  given  in  Table  38  is  probably  the 
simplest  and  most  direct. 


TABLE  38. — TABLES  SHOWING  MIXTURES  OP  TEST  MORTARS 
Test  Series  No.  1.    Cement  Content — 1  G.:  13  Sq.  In. 


Sand  letter 

Surface  area  per 
1000  g.,  sq.  in. 

Cement,  g 

Water,  cc. 

Ratio  of  cement 
to  aggregate  by 
weight 

A           

5,856.6 

450.5 

128.0 

1:2.22 

B               

5,106.1 

392.0 

111.5 

:2.55 

C 

7,683  .  7 

591.0 

134.5 

:    .69 

D  
E           

6,758.4 
12,816.4 

520.0 
986.0 

148.0 
280.5 

:    .92 
:    .12 

F      

6,769.1 

521.0 

148.0 

:    .92 

G        

4,182.0 

321.5 

91.5 

:    .11 

H           

6,564.6 

505.0 

143.5 

:    .98 

I.. 

6,564.6 

505.0 

143.5 

:    .98 

Test  Series  No.  2.     Cement  Content— 1  G.:  10,  15,  20  and  25  Sq.  In. 


F      

6,769 
6,769 

677.0 
451.0 

183.0 
132.5 

1:1.47 
1:2.21 

6,769 
6,769 

338.5 
270.5 

105.5 
92.5 

:2.95 
1:3.61 

CONCRETE  CONSTRUCTION  219 

Some  of  the  mathematical  relations  involved  are  of  interest. 
The  following  expression  shows  the  relation  between  the  fineness 
modulus  and  the  size  of  the  particle: 

m  =  7.94  +  3.32  log  d 

Where  m  =  fineness  modulus 

d  =  diameter  of  particle  in  inches 

This  relation  is  perfectly  general  so  long  as  we  use  the  standard 
set  of  sieves  mentioned  above.  The  constants  are  fixed  by  the 
particular  sizes  of  sieves  used  and  the  units  of  measure.  Loga- 
rithms are  to  the  base  10. 

This  relation  applies  to  a  single-size  material  or  to  a  given 
particle.  The  fineness  modulus  is  then  a  logarithmic  function 
of  the  diameter  of  the  particle.  This  formula  need  not  be  used 
with  a  graded  material,  since  the  value  can  be  secured  more 
easily  and  directly  by  the  method  used  in  Table  36.  It  is  appli- 
cable to  graded  materials  provided  the  relative  quantities  of  each 
size  are  considered,  and  the  diameter  of  each  group  is  used.  By 
applying  the  formula  to  a  graded  material  we  would  be  calculating 
the  values  of  the  separate  elemental  rectangles  shown  in  Fig. 
120. 

Proportioning  Concrete  by  Surface  Areas  of  Aggregates.1 — 
Volumetric  proportioning  of  concrete  is  notoriously  unsatis- 
factory. Many  investigators  have  been  studying  other  propor- 
tioning methods  which  will  at  the  same  time  be  practical  and  will 
insure  a  maximum  strength  of  concrete  with  any  given  material. 
The  latest  of  such  methods  and  one  which  in  the  tests  gives 
promise  of  some  success  is  that  devised  by  Capt.  L.  N.  Edwards, 
U.S.E.R.,  testing  engineer  of  the  Department  of  Works,  Toronto, 
Ontario,  which  was  explained  in  some  detail  in  a  paper  entitled 
1  Proportioning  the  Materials  of  Mortars  and  Concrete  by  Sur- 
face Areas  of  Aggregates,"  presented  to  the  American  Society  for 
Testing  Materials  at  its  annual  meeting  in  June. 

Briefly,  Captain  Edwards'  principle  is  that  the  strength  of  mor- 
tar is  primarily  dependent  upon  the  character  of  the  bond  exist- 
ing between  the  individual  particles  of  the  sand  aggregate,  and 
that  upon  the  total  surface  area  of  these  particles  depends  the 
quantity  of  cementing  material.  Reduced  to  practical  terms, 
this  means  that  a  mixture  of  mortar  for  optimum  strength  is  a 

lEnginesring  News-Record,  Aug.  15,  1918,  p.  317  et  seq. 


220  RETAINING  WALLS 

function  of  the  ratio  of  the  cement  content  to  the  total  surface 
area  of  the  aggregate  regardless  of  the  volumetric  or  weight 
ratios  of  the  two  component  materials.  As  a  corollary  to  his 
investigations,  Captain  Edwards  also  lays  down  the  principle 
that  the  amount  of  water  required  to  produce  a  normal  uniform 
consistency  of  mortar  is  a  function  of  the  cement  and  of  the  sur- 
face area  of  the  particles  of  the  sand  aggregate  to  be  wetted. 
Some  of  the  tests  deduce  the  fact,  already  demonstrated  in  a 
number  of  previous  tests,  that  strength  of  mortars  and  concrete 
is  a  definite  function  of  the  amount  of  water  used  in  the  mix. 

In  demonstrating  the  cement-surface  area  relation,  the  test 
procedure  was  as  follows :  First,  a  number  of  different  sands  were 
graded  through  nine  sieves,  varying  from  4  to  100  meshes  per 
inch,  and  the  material  passing  one  sieve  and  retained  on  the  next 
lower  was  separated  into  groups.  From  each  group,  then,  an 
actual  count  was  made  of  the  average  number  of  particles  of  sand 
per  gram.  For  the  larger  sizes  8  to  10  grams  or  more,  medium 
sizes  3  to  5  grams,  and  for  the  smallest  sizes  Y±  to  1  gram  were 
counted.  For  six  sands  counted,  including  a  standard  Ottawa 
which  is  composed  of  grams  passing  a  20  and  retained  on  a  30- 
mesh  sieve,  the  following  averages  were  obtained  for  the  number 
of  sand  particles  per  gram: 

Passing    4,  retained  on      8 14 

Passing    8,  retained  on    10 55 

Passing  10,  retained  on    20 350 

Passing  20,  retained  on    30 1,500 

Passing  30,  retained  on    40 4,800 

Passing  40,  retained  on    50 16,000 

Passing  50,  retained  on    80 40,000 

Passing  80,  retained  on  100 99,000 

With  a  specific  gravity  of  sand  of  2.689,  which  had  been  deter- 
mined by  a  number  of  tests,  the  average  volume  per  particle  of 
sand  was  determined  for  each  group,  and  assuming  that  the  shape 
of  the  particles  of  sand  was  spherical,  which  is  approximately 
correct,  the  surface  area  per  gram  of  sand  was  determined  for 
each  group.  The  results  are  shown  in  Fig.  121.  This  gave  a 
basis  of  surface  areas  for  the  various  groups  of  sand  in  hand. 

The  sands  were  then  regarded  to  different  granulometric 
analyses  in  order  to  get  representative  and  different  kinds  of 
aggregate  for  the  tests.  Using  these  sands  for  the  aggregate, 
numerous  briquets  and  cylinders  were  made  up  and  tested  in 


CONCRETE  CONSTRUCTION 


221 


tension  and  in  compression,  varying  the  mix  according  to  the 
ratio  of  the  weight  of  cement  to  the  surface  area  of  the  sand 
aggregate.  The  basis  of  the  ratio  of  grams  of  cement  to  square 
inches  of  surface  area  were  1 :10,  1 :15,  1 : 20  and  1 : 25.  The  con- 
sistency throughout  was  controlled  so  that  the  water  content 
would  not  affect  the  relative  strengths  of  the  different  specimens. 


20  40  00  60  100  120 

Diameter  of  Particle  of  Sand  in  0.001  Inch 

FIG.  121. — Capt.  Edwards'  method  of  surface  areas.     (From  Engineering  News- 
Record,  Aug.  15,  1918,  p.  317.) 

Test  mortars  were  then  made,  first,  by  keeping  the  cement- 
surface  area  ratio  constant  and  varying  the  kinds  of  sand;  second, 
by  varying  the  ratio  and  using  the  same  and.  These  two  series 
are  shown  in  the  accompanying  table.  As  will  be  noted  from 
Table  38,  in  test  series  No.  1  the  cement  content  is  one  gram  for 
thirteen  square  inches  of  surface  area,  but  the  sand  has  such  a 
different  grading  and  therefore  total  surface  area  that  the  ratio 
o  cement  to  aggregate  by  weight  varys  froml:1.12tol:3.11.  In 
spite  of  this  wide  variation  in  weight  and  therefore  in  volumetric 
relation  of  the  cement  to  the  aggregate,  the  strength  values,  as 
shown  in  Fig.  122,  were  markedly  constant.  In  series  No.  2 
the  cement  constant  varied  from  1  gram  to  10  sq.  in.  to  1  gram  to 
25  sq.  in.  of  sand  surface,  and,  as  shown  in  Fig  123,  the  strength 
curves  are  proportionat3  te  the  cement-area  ratio. 

Further  tests  were  made  by  Captain  Edwards  extending  this 
investigation  to  concrete,  and  while  these  showed  the  same  gen- 


222 


RETAINING  WALLS 


eral  results,  the  tests  were  not  sufficiently  elaborate  to  warrant  an 
abstract  of  them  here. 

It  might  seem  offhand  that  there  is  no  practical  occupation  to 
the  method.  Certainly,  the  very  considerable  labor  involved 
in  counting  125,000  sand  grams  for  one  sieve  group  alone  would 
deter  anyone  from  contemplating  such  a  program  for  practical 


5500 


300 


C        D       E        F 
5cmd  Le-tt-er 


FIG.  122.— Capt.  Edwards'  method  of  surface  areas.     (From  Engineering  News- 
Record,  Aug.  15,  1918,  p.  317.) 

work,  if  such  a  count  had  to  be  made  very  often.  However, 
Captain  Edwards  points  out  that  this  elaborate  counting  is 
required  only  as  a  preliminary  to  his  method  and  once  done  need 
not  be  repeated.  He  says: 

"The  adaptation  of  the  surface  area  method  of  proportioning  mortars 
and  concretes  to  both  laboratory  investigation  and  field  construction 


CONCRETE  CONSTRUCTION 


223 


operation  presents  no  serious  difficulty.  The  outstanding  feature  of 
this  method,  insofar  as  its  practical  application  is  concerned,  is  the  im- 
portance of  knowing  the  granulometric  composition  of  the  aggregate. 
The  securing  of  this  all  important  information  involves  a  comparatively 
small  amount  of  labor  and  by  way  of  equipment  the  use  of  only  the  nec- 
essary scales,  standard  sieves  and  screens.  The  time  element  involved 
is  comparatively  negligible,  since  the  computation  work  of  determining 
areas  and  quantities  of  cement  may  be  largely  reduced  to  the  most 
simple  mathematical  operation  by  the  use  of  tables  and  diagrams." 


5500 


4-500 


1* 


3500 


EL   8500 


1500 


10  20  30 

Surface  Area  per6ram  of 
Cement  in&q.In. 


10  20  30 

Surface  Area  per<5ram 
of  Ce  m  en+ i  n  Sq.  In . 


FIG.  123. — Capt.  Edwards'  method  of  surface  areas.     (From  Engineering  News- 
Record,  Aug.  15,  1918,  p.  317.) 

Diagrams  for  Laboratory  and  Field  Use. — For  use  in  the  labora- 
tory and  in  the  field,  diagrams  drawn  to  a  large  scale  increase 
accuracy  and  reduce  labor.  Fig.  124  is  designed  for  use  in  de- 
termining the  surface  area  of  sand  aggregate.  It  is  intended  for 
laboratory  use.  Fig.  125  is  the  same  sort  of  diagram  intended 
for  both  laboratory  and  field  use.  The  diagrams  are  derived 
from  information  obtained  in  the  tests.  Fig.  126  is  designed 
for  use  in  determining  the  surface  of  stone  aggregate,  and  is  in- 
tended for  both  field  and  laboratory  use,  and  Fig.  127  shows  the 
conversion  diagram  for  determining  the  relative  quantity  of 
cement  in  pounds  per  100  Ib.  of  sand,  and  the  corresponding 
relation  of  cement  in  grams  to  the  surface  area  of  1,000  grams  of 
sand,  and  vice  versa.  The  author  then  gives  the  following  ex- 
ample of  how  the  diagrams  shown  in  Figs.  124-127  may  be  used: 


224 


RETAINING  WALLS 


CONCRETE  CONSTRUCTION 


225 


160 


FIG.  127.- 


5          <o          7          S         9         10        II         \2        13        14- 
Surfoice  Area  of  lOOOg.  of  Sand;  Thousand  Sq.In. 

-(From  Engineering  hews-Record,  Aug.  15,  1918.     Capt.  Edwards.) 


Example  No.  1. — Required  to  find  the  composition  of  a  batch 
of  mortar  using  1,000  g.  of  sand  A  and  a  cement  content  propor- 
tioned: 1  g.  cement  to  15  sq.  in.  sand  area. 


SAND  AREA 


Sieve 
P    4-R 

8  

P    8-R 

10       . 

P  10-R 

20 

P  20-R 

30..    . 

P  30-R 

40 

P  40-R 

50  

P50-R 

80.. 

P  80-R  100.. 


Totals . . 


Grading, 
per  cent. 

15.0 

5.0 

25.0 

15.0 

15.0 

10.0 

10.0 

5.0 

100.0 


Weight,  g. 

Area  (Fig.  4), 
sq.  in. 

150 

142 

50 

75 

250 

694 

150 

676 

150 

997 

100 

992 

100 

1,348 

50 

932 

1,000 


5,856 


Cement  (g.)  =  =  390.5 

lo 


Water  (c.c.)  =  I  390.5  X  22.25  per  cent,  (normal  consistency)  f  + 


5856 
210 


=  115 


226  RETAINING  WALLS 

The  author  does  not  give  anywhere  what  he  considers  to  be 
proper  ratio  of  the  cement  to  the  sand  surface  area.  That  would 
presumably  have  to  be  determined  by  investigations  of  the  ag- 
gregates involved  in  any  case. 

Ratio  of  Fine  to  Coarse  Aggregate  Basis  for  Concrete  Mix- 
ture.1— Another  method  of  proportioning  concrete  mixtures  is 
proposed  by  R.  W.  Crum,  in  a  paper,  read  before  the  American 
Society  for  Testing  Materials,  and  entitled  "  Proportioning  of 
Pit-Run  Gravel  for  Concrete."  The  method  was  devised  for 
and  is  specially  applicable  to  Middle  Western  gravels  which  occur 
in  assorted  gradings.  By  its  use  a  proper  concrete  can  be  had 
with  any  pit  gravel  by  the  addition  of  the  correct  amount  of 
cement,  to  be  determined  by  the  method.  Basically,  the  author's 
scheme  rests  on  the  assumption  that  the  ratio  of  cement  to  air 
and  water  voids  is  an  indication  of  strength.  In  other  words, 
the  nearer  the  cement  content  approaches  the  volume  of  the 
voids  the  greater  is  the  strength  of  the  concrete.  He  assumes 
that  for  certain  classes  of  concrete — that  is,  for  concrete  to  be 
used  under  certain  conditions — there  is  an  optimum  sand-aggre- 
gate ratio.  In  that  ideal  mix  the  cement-void  ratio  is  computed 
and  the  amount  of  cement  necessary  to  bring  the  actual  mix  up  to 
that  ratio  is  found.  This  gives  the  best  mixture — reducing  to 
loose  volume — for  that  particular  aggregate.  Although  the  au- 
thor states  that  the  proper  grading  depends  upon  the  consistency 
or  amount  of  water  in  the  mixture,  and  although  he  says  specifi- 
cally that  one  must  get  a  concrete  which  will  yield  a  workable 
mixture  for  the  conditions  under  which  it  is  placed,  he  does  not 
tell  in  the  paper  just  what  degree  of  workability  is  reached  by  his 
method  nor  the  standard  of  consistency  or  workability  which  was 
used  in  making  the  tests.  He  claims  that  the  method  gives 
results  about  midway  between  the  fineness-modulus  method  of 
Abrams  and  the  surface-area  method  of  Edwards.  Analyses  of 
prospective  aggregates  may  be  readily  made  in  the  field  for  the 
method,  inasmuch  as  it  requires  only  to  be  known  the  gradings 
above  and  below  a  No.  4  sieve. 

1  Engineering  News-Record,  July  10,  1919. 


PLATE  V 


FIG.  A. — Method  of  laying  stone  wall  by  series  of  derricks. 

(Facing  page  227) 


PLATE  VI 


FIG    A. —  Uncoursed  rubble  wall  with  coursed  effect  given  by  false  pointing. 


Fia.  B. — Rubble  wall  (Los  Angeles)  with  face  formed  by  nigger-heads. 


CHAPTER  IX 
WALLS  OTHER  THAN  CONCRETE 

Plant. — Rubble  and  cut-stone  walls  up  to  5  or  6  feet  in  height 
are  built  of  stone  of  such  size  that  they  are  easily  raised  and  set 
by  hand.  No  special  plant  is  therefore  required  and  the  wall  is 
built  entirely  by  hand  labor.  As  the  walls  increase  in  height, 
good  construction  requires  the  use  of  larger  stone,  to  insure  a 
wall  properly  bonded  together  and  it  becomes  necessary  then 
to  employ  plant  to  raise  and  set  the  large  stone.  A  derrick, 
either  a  guy  or  a  stiffleg,  is  probably  the  most  serviceable  and 
efficient  piece  of  plant  to  use  in  setting  stone  walls.  It  is  op- 
erated by  a  hoist  run  by  steam,  electricity  or  air.  A  guy  derrick 
is  possibly  preferable  in  that  it  permits  a  greater  swing  of  the 
boom.  It  is  limited  however  by  the  fact  that  it  requires  ample 
room  to  anchor  its  guys,  room  not  always  available,  especially 
in  city  work.  A  stiffleg  derrick  is  a  self  contained  unit,  the 
weight  of  the  hoist  and  power  plant  providing  the  necessary 
anchorage. 

In  setting  a  derrick  care  should  be  observed  that  it  is  placed 
back  from  the  wall  a  distance  sufficient  to  ensure  'topping '  out 
the  wall.  When  the  yardage  of  masonry  permits,  it  is  most 
economical  and  proves  most  time  saving  to  set  up  a  chain  of 
derricks  at  such  intervals  that  no  gaps  are  left  in  the  wall.  This 
continuity  of  the  work  will  obviate  the  tendency  to  cracks  caused 
by  joining  up  new  work  with  old  work  (see  Chapter  V,  "  Settle- 
ment")- The  derricks,  when  set  up  in  sequence,  are  easily  dis- 
mantled and  set  up  in  their  new  positions  by  aid  of  the  adjoining 
derricks.  Photographic  Plate  No.  V,  Fig.  A  shows  the  method 
of  constructing  high  rubble  walls  (over  32  feet  high)  by  means 
of  such  plant. 

Mortar. — The  mortar  for  use  in  the  rubble  masonry  walls  is 
mixed  alongside  the  wall  and  is  delivered  to  the  working  gangs 
in  bucket  loads  as  required.  The  usual  mortar  is  mixed  in  pro- 
portions^  of  one  cement  to  three  sand.  For  work  of  large  size 

227 


228  RETAINING  WALLS 

conveniently  located,  it  may  prove  economical  to  mix  the  mortar 
by  machine  in  a  central  plant  and  deliver  by  cart  or  otherwise 
over  the  work.  Usually,  however,  it  has  proven  most  efficient 
to  mix  the  mortar  by  hand  for  each  gang,  or  for  two  adjacent 
gangs.  The  cement  required  for  a  rubble  masonry  wall  of 
fairly  large  size  (varying  from  twelve  to  forty  feet  in  height) 
will  average  about  one  and  one-half  bags  to  the  finished  yard  of 
wall.  Due  care  in  dressing  the  stone  and  chinking  up  the 
interstices  with  spalls  will  help  to  keep  the  amount  of  cement 
required  to  a  minimum.  With  mortar  mixed  in  the  proportion 
of  one  cement  to  three  sand,  the  finished  wall  should  contain  from 
15  to  20  per  cent,  mortar. 

Construction  of  Wall. — In  constructing  the  wall  the  largest 
stone  should  be  placed  at  the  bottom  course.  If  the  soil  is  a 
slightly  yeilding  one  the  stones  may  be  dropped  from  a  height 
of  two  to  three  feet  to  insure  their  thorough  imbedment.  The 
bottom  course  may  consist  of  a  lean  concrete  in  place  of  the 
rubble  stone.  The  wall  should  have  a  proper  proportion  of 
headers  (stones  lying  transversely)  usually  about  J£  of  the  total 
yardage.  The  stones  should  be  most  carefully  bedded,  and  all 
the  interstices  filled  with  spalls  and  if  the  wall  is  a  mortar  one, 
finished  off  with  the  cement  mortar.  The  construction  of  a 
rubble  masonry  wall,  both  dry  and  cement  requires  a  most 
conscientious  cooperation  between  the  engineer  and  the  con- 
tractor and  it  is  only  by  such  mutual  aid  that  a  good  masonry 
wall  can  be  built.  When  a  section  of  wall  is  to  be  finished  some 
time  before  the  adjoining  section  is  to  be  built  it  is  well  to  "rack" 
back  the  sides  to  insure  a  good  bond  between  the  old  and  new 
work.  It  must  be  remembered  that  a  masonry  wall  has  no 
expansion  joints  and  that  all  movement  of  the  wall,  must  be 
taken  up  by  the  masonry  itself.  Cracks  will  therefore  a  pear 
along  the  plane  of  weakness  and  unless  great  care  has  been 
exercised  in  the  laying  of  the  wall,  these  cracks  will  become  very 
disfiguring. 

The  stone  should  be  good,  sound  stone,  thoroughly  cleaned 
and  roughly  dressed  to  take  off  the  soft  and  cracked  edges.  It 
should  be  wet  before  setting,  especially  in  hot  weather.  Friable 
and  soft  stone  should  not  be  used.  An  excellent  example  of 
rubble  masonry  specifications,  is  quoted  here.1 

1  Track  Elevation,  Philadelphia,  Germantown  and  Norristown  R.  R., 
S.  T.  WAGNER,  Trans.  A.S.C.E.,  Vol.  Ixxvi,  p.  1833. 


WALLS  OTHER  THAN  CONCRETE  229 

"  Third-class  masonry  shall  be  formed  of  approved  quarry  stone  of  good 
shape  and  of  good  flat  beds.  No  stone  shall  be  used  in  the  face  of  the 
walls  less  than  6  inches  thick  or  less  than  12  inches  in  their  least  hori- 
zontal dimension. 

Headers  shall  generally  form  about  %  of  the  faces  and  backs  of  the 
walls  with  a  similar  proportion  throughout  the  mass  when  they  do  not 
interlock,  and  the  face  stones  shall  be  well  scabbed  or  otherwise  worked 
so  that  they  may  be  set  close  and  chinking  with  small  stone  avoided. 

In  walls  five  feet  thick  or  less,  the  stones  used  shall  average  6  to  8 
cubic  feet  in  volume  and  the  length  of  the  headers  shall  be  equal  to  two- 
thirds  the  thickness  of  the  wall.  In  walls  more  than  five  feet  in  thick- 
ness the  stones  used  shall  average  12  cubic  feet  in  volume  and  the  headers 
shall  not  be  less  than  four  feet  long.  Generally  no  stones  shall  be  used 
having  a  less  volume  than  four  cubic  feet  except  for  filling  the  interstices 
between  the  large  stones. 

In  no  case  shall  stones  be  used  having  a  greater  height  or  build  than 
30  inches  and  these  stones  must  bond  the  joints  above  and  below  at  least 
18  inches;  in  all  other  cases  the  smaller  stone  must  bond  the  joints  above 
and  below  at  least  10  inches. 

The  stones  in  the  foundation  shall  generally  not  be  less  than  10  inches 
in  thickness  and  contain  not  less  than  10  square  feet  of  surface.  The 
foundation  shall  consist  of  1:3:6  concrete,  if  so  directed  by  the  Chief 
Engineer." 


Coping. — The  wall,  either  dry  or  cement,  is  usually  topped 
with  a  coping.  Expansion  joints  in  this  coping  should  be 
placed  at  intervals  of  about  five  to  ten  feet.  The  sections  may 
be  separated  by  plain  paper  or  may  be  tarred.  The  coping 
should  preferably  be  placed  after  the  wall  has  been  constructed 
for  some  time.  This  permits  settlement  to  take  place  and  where 
definite  cracks  appear  in  the  wall,  expansion  joints  may  be  placed, 
to  avoid  unsightly  cracking  of  the  coping  itself.  When  built  of 
concrete,  the  coping  should  be  about  one  foot  thick  and  offset 
from  the  face  of  the  wall  about  3  or  4  inches.  The  form  for 
the  coping  should  be  well  built  and  carefully  lined.  Any  care- 
lessness in  lining  the  coping  forms  shows  in  a  wavy  broken  coping 
line  and  proves  unsightly.  The  forms  should  be  built  of  2  inch 
stock,  carefully  wired  and  braced.  This  will  prevent  the  bulging 
of  the  coping  face  and  the  thickness  of  the  form  will  permit  a 
frequent  reuse  of  the  form. 

If  a  stone  coping  is  desired,  a  blue  stone  flagging  from  4  to 
8  inches  in  thickness  makes  an  effective  top  finish  for  the  wall 


230  RETAINING  WALLS 

Face  Finish. — The  face  of  a  rubble  or  other  masonry  wall, 
receives  such  treatment  as  the  environment  of  the  wall  requires 
(see  Chapter  X  on  " Architectural  Treatment").  With  care 
in  the  selection  of  face  stone  and  with  a  fair  attempt  to  dress 
these  stones,  the  wall  needs  but  little  other  work  upon  it  except 
some  pointing  of  the  joints.  As  the  demand  for  special  face 
treatment  increases,  more  attention  must  be  paid  to  the  selection 
of  face  stones  and  to  the  pointing  of  the  joints.  Face  treatment 
may,  roughly  be  divided  into  the  following  classifications :  Rough 
pointing;  special  or  false  pointing;  selection  of  special  face  stone; 
plaster  finishes. 

Rough  Pointing. — After  laying  the  wall,  the  stones  are  cleaned 
of  whatever  mortar  has  accidentally  dropped  upon  them.  The 
joints  are  raked  and  then  brought  to  the  rough  face  plane  with 
mortar.  For  walls  as  generally  built  in  the  outlying  districts, 
this  type  of  treatment  is  sufficient. 

False  Pointing. — To  obtain  a  somewhat  more  pleasing  and 
decorative  effect,  rough,  uncoursed  masonry  is  pointed  falsely, 
to  give  the  appearance  of  coursed  masonry.  After  cleaning  the 
face  stones  the  face  of  the  wall  is  brought  to  a  rough  plane  and  is 
then  coursed  with  the  trowel  into  rectangles.  Work  of  this 
nature  is  not  of  great  permanence,  the  mortar  slowly  spalling 
off  with  the  weather.  (To  secure  the  coursed  masonry  effect, 
more  surface  of  the  wall  must  receive  a  mortar  coat  than  is 
necessary  otherwise.)  It  is  of  questionable  taste  to  attempt 
to  mask  the  nature  of  the  wall  by  such  face  treatment.  This 
mode  of  treatment  is  usually  limited  to  small  walls  forming  the 
street  walls  of  residential  plots.  A  photograph  of  this  class  of 
wall  is  shown  here  (Photo  Plate  VI,  Fig.  A). 

Special  Stone. — The  character  of  the  masonry  comprising  the 
wall  body  may  be  completely  masked  by  forming  the  face  of  the 
wall  with  specially  selected  stone.  The  rough  masonry  may  then 
be  considered  a  backing  for  the  selected  stone  masonry.  For 
walls  entering  into  a  costly  and  decorative  scheme  of  landscape 
work,  the  face  may  be  made  an  ashlar,  or  other  coursed  masonry 
effect,  using  limestone,  sandstone  or  granite.  When  the  walls 
are  of  considerable  thickness  it  is  usual  to  build  them  thus,  with 
the  expensive  stone  at  the  surface  only.  Walls  of  this  type  are 
the  most  costly  of  all  walls,  yet  present  the  most  imposing  and 
pleasing  types  of  masonry  construction.  The  details  of  construc- 
tion of  these  walls  are  thoroughly  discussed  in  a  number  of 


WALLS  OTHER  THAN  CONCRETE  231 

standard  text-books  (e.g.  Baker's  "Masonry  Construction") 
and  need  not  be  mentioned  here. 

A  very  pleasing  effect  secured  by  the  use  of  boulders  or  "  nigger 
heads  "  is  shown  here  (see  Photo  Plate  VI,  Fig.  B)  (used  extensively 
in  Los  'Angeles).  Various  modifications  of  work  of  this  kind 
are  readily  adapted  to  local  environments  with  exceptionally 
pleasing  results. 

Plaster  Coats. — This  is  probably  the  least  desirable  of  surface 
finishes,  both  in  effect  and  in  duration  of  life.  Because  of  its 
limited  permanence  great  care  must  be  exercised  in  applying 
these  coats  to  the  face  of  rough  masonry  walls.  Plaster  or 
stucco  coats,  when  applied  to  the  face  of  a  wall,  are  rough  cast 
or  stippled.  No  trowelling  is  done  upon  the  face,  the  mortar 
being  placed  with  the  usual  wooden  mortar  board.  To  insure 
permanence  some  form  of  wire  mesh  or  other  netting  should  be 
fastened  to  the  face  of  the  wall  to  hold  the  plaster  coat.  The 
netting  may  be  attached  to  wooden  plugs  inserted  in  the  mortar 
while  the  wall  is  in  the  course  of  construction. 

Cost  Data. — The  following  is  an  analysis  of  the  cost  of  a  wall 
36  feet  high,  averaging  about  13  cubic  yards  to  the  running  foot. 
It  is  merely  a  labor  charge  and  does  not  include  the  cost  of  obtain- 
ing the  stone,  etc. 

CEMENT    RUBBLE  WALL.     2750  CUBIC  YARDS 

Foreman,  114  days  at  $6.00  per  day $684.00 

Masons,  167  days  at  $4.50  per  day 751 . 50 

Hoistrunner,  113  days  at  $6.00  per  day 678 . 00 

Signalman,  90  days  at  $2.50  per  day 225.00 

Laborers,  625  days  at  $2.50  per  day 1562. 50 

Total  cost $3901 .00 

The  average  cost  per  yard,  exclusive  of  all  overhead,  insurance,  plant 
charges,  materials,  etc.,  is  $1.42  per  yard. 


CHAPTER  X 
ARCHITECTURAL  DETAILS,    DRAINAGE,  WATERPROOFING 

Architectural  Treatment. — Concrete  retaining  walls  form  a 
class  of  engineering  structures  for  which  ornate  decorations  are 
of  questioned  taste.  Occasionally,  however,  some  special  face 
treatment  becomes  necessary  to  permit  the  wall  to  enter  into 
the  general  landscape  improvement  involving  a  particular  archi- 
tectural scheme.  Thus,  for  example,  retaining  walls  forming  an 
approach  to  a  bridge,  especially  a  concrete  arch  are  usually 
made  to  follow  the  general  viaduct  architecture.  Walls  for  a 
railroad  station,  where  the  main  line  is  on  the  fill,  must  be  in 
keeping  with  the  architectural  motive  of  the  building  itself. 
Walls  in  parks  must  receive  such  treatment  as  will  make  them 
harmonize  with  the  park  landscape  work.  In  general,  however, 
simplicity  of  treatment  is  essential,  to  conform  with  good  taste. 

Concrete  walls  are  finished  on  top  with  a  coping;  usually  about 
one  foot  thick  and  projecting  3  to  6  inches  beyond  the  face  of 
the  wall.  In  addition  a  hand  rail,  picket  fence,  or  concrete 
parapet  wall  is  placed  on  top  of  the  wall  of  plain  or  ornamental 
effect  as  conditions  indicate.  The  face  of  the  wall  receives  such 
treatment  as  will  remove  the  unavoidable  blemishes  of 
construction. 

Face  Treatment.— The  concrete  face  of  the  retaining  wall  may 
either  be  rubbed,  tooled  or  receive  a  special  composition  surface. 
Preliminary  to  applying  the  face  treatment,  the  tie  rods,  wires, 
etc.  are  cut  back,  and  the  face  patched  where  necessary,  employ- 
ing a  stiff  mortar  for  this  purpose.  To  insure  a  successful  surface 
finish,  it  is  imperative  that  the  wall  be  well  built.  A  surface 
finish  cannot  conceal  poor  work  and  poor  work  will  eventually 
destroy  the  best  surface  finish.  The  less  a  wall  is  patched  or 
otherwise  repaired,  the  more  certain  it  is  that  the  surface  treat- 
ment will  be  of  pleasing  and  permanent  character.  Board  marks 
are  left  after  the  forms  are  stripped  which  may  be  more  or  less 
masked  by  careful  treatment.  It  may  be  set  down  as  almost 
axiomatic  that  board  marks  can  never  be  entirely  eradicated, 

232 


ARCHITECTURAL  DETAILS  233 

no  matter  what  face  treatment  is  applied.  For  this  reason  care 
must  be  taken  in  the  continued  use  of  the  same  set  of  forms,  so 
that  no  panel  is  used  in  the  face  after  its  edges  become  splintered 
or  frayed. 

It  has  been  pointed  out  in  a  previous  chapter  that  construction 
joints  leave  a  distinct  cleavage  mark.  To  make  sure,  for  walls 
that  will  occupy  a  position  of  more  or  less  architectural  promi- 
nence, that  there  shall  be  no  construction  joints,  it  is  specified 
that  the  section  of  wall  between  the  expansion  joints  shall  be 
poured  completely  in  one  operation.  This  is  a  praiseworthy 
mandate  and  is  worthy  of  adoption  for  all  character  of  work, 
regardless  of  merely  the  insistence  of  an  architectural  finish. 
The  distance  between  expansion  joints  may  be  made  such  that  it- 
is  practicable  to  pour  a  section  complete  with  ordinary  plant  in 
one  pour. 

Defective  concrete  work  appearing  at  the  surface  must  be 
removed  immediately  upon  stripping  the  forms  and  a  rich  mortar 
concrete  inserted.  Haphazard  patchwork  will  not  do.  It  is 
but  a  temporary  expedient  and  the  patch  will  soon  spall  off 
leaving  a  disfigured  wall.  A  photograph  of  a  wall  so  treated  is 
shown  here  (See  Fig.  A,  Plate  VII)  and  is  eloquent  of  the  effects 
of  poor  concrete  work  and  poor  patch  work. 

If  the  forms  are  not  held  tight,  or  are  not  carefully  caulked 
above  work  already  completed,  the  yielding  of  the  form,  even  to  a 
minute  degree,  will  permit  the  grout  to  run  down  coating  and 
disfiguring  the  concrete  work. 

Briefly  stated,  conscientious  vigilance  in  the  observance  of  the 
edicts  of  good  concrete  work  is  the  price  of  a  good  surface  finish 
and  using  the  analogy  of  pathology,  diseases  of  the  concrete 
body  of  a  wall  are  usually  exhibited  by  symptoms  of  facial 
blemishes. 

Rubbing. — The  face  of  concrete  mirrors  most  faithfully  the 
inside  face  of  the  form,  bringing  out  the  delineations  of  the  board 
marks,  the  lips  of  the  panels,  etc.  Immediately  upon  stripping 
the  forms,  and  after  cutting  the  rods  and  wires  where  necessary, 
and  after  making  such  patches  as  are  indicated,  the  face  is  rubbed 
down  with  an  emery  block,  and  a  thin  grout  wash  is  applied  at 
the  same  time.  The  fresher  the  concrete,  the  easier  it  is  to 
remove  the  facial  blemishes  by  rubbing  and  it  is  therefore  im- 
perative that  the  forms  be  stripped  as  soon  as  good  construc- 
tion permits.  For  the  average  environment,  and  over  90  per  cent. 


234  RETAINING  WALLS 

of  retaining  walls  are  built  in  such  environment,  rubbing  a  wall 
presents  finally  a  surface  that  is  sufficiently  pleasing. 

In  applying  the  grout  wash,  care  must  be  taken  to  use  a  con- 
stant proportion  of  the  cement  and  water.  It  is  quite  possible, 
where  the  rubbing  is  not  done  on  one  day,  to  use  grout  mixes  of 
different  strengths  leaving  the  surface  finished  in  two  shades. 

Tooling. — If  the  cement  skin  of  a  concrete  wall  is  removed  by 
sharp  bits,  the  abraded  surface  gives  a  rough  stone  appearance 
quite  pleasing  in  effect.  This  skin  may  be  removed  by  hand  with 
an  ordinary  wedge  bit,  or  with  special  two,  four  and  six  edged 
bits.  If  there  is  a  large  amount  of  surface  to  be  so  treated,  it  is 
a  matter  of  economy  to  use  an  air  drill  to  work  the  hammer. 
The  hammer  is  passed  lightly  over  the  surface,  applied  just  long 
enough  to  remove  the  grout  skin,  care  being  taken  not  to  start 
ravelling  the  stone.  A  gravel  concrete  seems  to  give  a  better 
appearance  than  a  broken  stone  concrete,  the  sparkling  effect 
of  the  pebbles  presenting  an  excellent  appearance,  especially 
in  the  direct  sunlight.  When  broken  stone  is  used,  the  size  of  the 
stone  should  be  limited  to  %  inch  stone,  the  ordinary  commercial 
stone.  With  larger  stone  it  is  difficult,  in  tooling  the  wall  to 
prevent  ravelling. 

It  is  understood  that  tooling  is  much  more  expensive  than  rub- 
bing (roughly  about  ten  to  fifteen  times)  and,  ordinarily  is  only 
specified  to  effect  a  special  architectural  feature. 

As  in  the  case  of  rubbing,  the  concrete  wall  must  be  carefully 
patched  and  construction  devices,  such  as  rods,  wires,  etc.,  removed 
or  cut  back  several  inches  from  the  surface. 

It  is  usual  to  finish  the  edges  of  a  tooled  surface  by  means  of  a 
rubbed  border  of  one  or  more  inches  in  width.  Care  must  be 
taken  not  to  tool  too  near  an  edge  as  the  concrete  may  be  broken 
off. 

Special  Finishes. — To  enhance  the  architectural  appearance 
of  a  retaining  wall,  a  special  face  finish  is  applied  to  the  wall, 
masking  its  construction  finish.  An  ordinary  plaster  coat  may 
be  applied  to  the  wall,  or  a  granolithic  or  other  fine  grit  finish  may 
be  placed  upon  its  surface.  In  applying  such  a  coat  it  is  essential 
that  due  appreciation  should  be  had  of  the  proper  bond  between 
the  wall  and  the  coat.  To  apply  a  coat  of  mortar  or  other  finish 
after  the  forms  have  been  stripped  and  the  wall  set  gives  little 
assurance  of  a  permanent  finish.  The  coefficients  of  expansion 
between  the  wall  concrete  and  the  rich  mortar  are  unlike,  produc- 


ARCHITECTURAL  DETAILS  235 

ing  eventually  voids  between  the  wall  and  coat.  The  action  of 
frost  and  the  other  destructive  elements  finally  cause  the  coat  to 
spall.  It  is  therefore  usually  specified  that  the  finish  coat  shall  be 
applied  simultaneously  with  the  pouring  of  the  wall,  so  that  the 
coat  is  a  part  of  the*  wall  itself,  and  is  therefore  more  or  less  immune 
to  the  weathering  actions.  An  excellent  specification  for  a  grano- 
lithic coat  is  quoted  here  and  may  be  used  as  a  model  clause  for 
all  grit  finishes.1 

"  Surf  ace  of  concrete  exposed  to  the  street  shall  be  composed  of  one 
part  cement,  two  parts  coarse  sand  or  gravel  and  two  parts  granolithic 
grit,  made  into  a  stiff  mortar.  Granolithic  grit  shall  be  granite  or  trap 
rock  crushed  to  pass  a  %  inch  sieve  and  screened  of  dust.  For  vertical 
surfaces  the  mixture  shall  be  deposited  against  the  face  forms  to  a  least 
thickness  of  one  inch  by  skilled  workmen,  as  the  placing  of  concrete 
proceeds  and  thus  form  a  body  of  the  work.  Care  shall  be  taken  to 
prevent  the  occurrence  of  air  spaces  or  voids  in  the  surface.  The  face 
forms  shall  be  removed  as  soon  as  the  concrete  has  sufficiently  hardened 
and  any  voids  that  may  appear  shall  be  filled  up  with  the  mixture. 

"The  surface  shall  then  be  immediately  washed  with  water  until  the 
grit  is  exposed  and  rinsed  clean  and  protected  from  the  sun  and  kept 
moist  for  three  days.  For  horizontal  surfaces  the  granolithic  mixture 
shall  be  deposited  on  the  concrete  to  a  least  thickness  of  1.5  inches 
immediately  after  the  concrete  has  been  tamped  and  before  it  has  set 
and  shall  be  trowelled  to  an  even  surface  and  after  it  has  set  sufficiently 
hard  shall  be  washed  until  the  grit  is  exposed. 

"All  concrete  surfaces  exposed  to  the  street  shall  be  marked  off  into 
courses  in  such  detailed  manner  as  may  be  directed  by  the  Chief 
Engineer." 

Finishes  of  various  colors  may  be  secured  by  the  use  of  properly 
colored  grit.  A  red  finish  may  be  secured  by  the  use  of  brick 
grit;  a  gray  by  bluestone  screenings,  etc.  Below  is  a  method 
of  obtaining  still  another  type  of  surface  finish.2 

"A  surface  finish  for  concrete,  whereby  a  sand  coating  is  applied  may 
be  secured  by  the  following  method,  outlined  by  Mr.  Albert  Moyer 
of  the  Vulcanite  Portland  Cement  Co.  Erect  forms  of  rough  boards 
in  courses  of  three  feet  or  less  and  plaster  the  insides  with  wet  clay 
worked  to  a  plastic  consistency.  While  the  clay  is  wet  apply  evenly 
loose  buff,  red  or  other  colored  sand  and  then  pour  in  the  concrete. 

1  S.  T.  WAGNER,  Track  Elevation,  Philadelphia,  Germantown  and  Norris- 
town  Railroad,  Trans.  A.S.C.E.,  Vol.  Ixxvi,  p.  1836. 
3  Engineering  Record,  Vol.  61,  p.  454.    ' 


236 


RETAINING  WALLS 


After  removing  the  forms,  wash  off  the  clay  with  water  and  if  necessary 
scrub  lightly  with  a  brush.  The  sand,  Mr.  Moyer  states  will  adhere  to 
the  concrete  and  givs  a  surface  of  pleasing  color  and  texture. 

The  following  table  gives  the  proportion  of  coloring  matter 
to  use  to  secure  a  desired  shade  of  concrete  finish.  The  table 
is  taken  from  "Concrete  Construction  for  Rural  Communities/' 
by  Roy  A.  Seaton,  page  148. 


Color  of  hardened 
mortar 

Mineral  to  be  used 

Pounds  of  color 
to  each  bag  of 
cement 

Gray.  . 

Germantown  lamp-black 

u 

Black 

Manganese  dioxide 

12 

Black  

Excelsior  carbon  black       

3 

Blue. 

Ultramarine  blue 

5 

Green 

Ultramarine  green 

6 

Red  

Iron  oxide  

6 

Bright  red 

Pompeian  or  English  red     .   .        

6 

Brown  

Roasted  iron  oxide  or  brown  ochre  

6 

Buff 

Yellow  ochre 

6 

"Colors  will  usually  be  considerably  darker  while  the  concrete  is 
wet  than  after  it  dries  out  and  the  colors  are  likely  to  grow  somewhat 
lighter  with  age.  Hence  considerably  more  pigment  should  be  used 
than  is  necessary  to  bring  wet  concrete  or  mortar  to  the  desired  shade." 

Artistic  Treatment  of  Concrete  Surfaces  in  General. — The 

treatment  of  concrete  surfaces  of  all  types  is  ably  discussed  in  a 
book  by  Lewis  and  Chandler,  "  Popular  Hand  Book  for  Cement 
and  Concrete  Users "  (see  chapter  "Artistic  Treatment  of  Con- 
crete Surfaces")-  The  various  methods  of  finishing  a  concrete 
surface  are  classified  as  follows : 

"  1.  Spading  and  trowelling  the  surface. 

"2.  Facing  with  Stucco. 

"3.  Facing  with  Mortar. 

"4.  Grouting. 

"5.  Scrubbing  and  washing. 

"6.  Etching  with  Acid. 

"7.  Tooling  the  Surface  with  Bush-hammers  or  other  tools. 

"8.  Surfacing  with  gravel  or  pebbles. 

"9.  Tinting  the  surface. 

"10.  Panelling,  Mosaic,  carving  etc." 


PLATE  VII 


FIG.  A. — Showing  effects  of  poor  concrete  work. 


Fia.  B. — Ornamental  parapet  wall.     Tooted  with  rubbed  borders. 

(Facing  page  236) 


PLATE  VIII 


Fio.  A. — Ornamental  handrail — approach  to  viaduct. 


FIG.  B. — Picket    fence.     Wall    lining    open  cut  approach  to  depressed  street 

crossing. 

(Facing  page  236) 


PLATE  IX 


FIG.  A. — Ornamental  concrete  handrail  approach  to  concrete  arch. 


ARCHITECTURAL  DETAILS  237 

The  methods  specially  applicable  to  retaining  walls  have  been 
analyzed  in  detail  in  the  present  chapter. 

In  connection  with  the  artistic  treatment  of  retaining  wall 
surfaces,  it  may  prove  of  interest  to  note  that  an  exhaustive 
study  of  a  special  surface  was  made  by  John  J.  Earley,  Proceedings 
American  Concrete  Institute,  1918,  in  a  paper  entitled  "Some 
Problems  in  Devising  a  New  Finish  For  Concrete."  The  wall 
under  discussion  was  built  in  Meridian  Hill  Park,  Washington, 
D.C.  The  original  plans  called  for  a  stucco-finished  wall.  A 
sample  of  wall  with  such  a  finish  was  built.  "The  result  was  a 
plaster  wall,  nothing  more  **********  fae  wa}j  was 
without  scale.  It  did  not  give  the  appearance  of  strength  or 
size  equal  to  its  task  as  a  retaining  wall. "  It  was  finally  decided 
to  strip  the  forms  of  the  wall  as  soon  as  possible  after  pouring 
(from  24  to  48  hours)  and  scrub  the  surface  with  steel  brushes 
"until  the  aggregate  was  exposed  as  evenly  as  possible. " 

"This  method  of  treating  the  surfaces  at  once  supplied  the  sense  of 
strength  and  size  that  was  lacking  before.  The  wall  was  no  longer  a 
plastered  one,  but  was  reinforced  concrete  and  nothing  else,  and  it 
seemed  big  and  strong  enough  to  suit  all  demands  that  would  be  made 
upon  it." 

The  face  was  panelled  and  the  piers  were  treated  differently, 
to  afford  a  contrast  to  the  tooled  surfaces. 

Hand  Rails. — To  prevent  accidents  and  trespassing  or  to  lend 
a  pleasing  finish  to  a  retaining  wall  a  railing  of  some  kind  is 
built  into  the  coping  of  the  wall,  of  a  character  in  conformity 
with  the  needs  of  the  environment.  When  a  wall  retains  an  em- 
bankment rising  above  the  surrounding  country,  the  railing  is 
required  as  a  protection  to  those  walking  along  the  edge  of  the  em- 
bankment. If  the  environment  demands  a  railing  more  ornate  in 
character,  the  railing  may  be  made  of  concrete,  stone,  concrete 
blocks,  etc.  Some  photographs  of  railings  of  this  latter  character 
are  shown  here  (see  Plates  VII,  Fig.  B,  VIII,  Fig.  A  and  IX,  Fig.  A). 
To  prevent  trespassing,  by  climbing  over  low  walls,  or  walls  which 
line  cuts  along  a  highway,  it  is  usual  to  build  a  picket  fence. 
A  photograph  of  a  standard  type  of  such  fence  is  shown  on  Plate 
VIII,  Fig.  B. 

The  metal  railings  are  anchored  to  the  wall  by  bolts.  Holes 
are  drilled  in  the  wall  coping  to  fit  the  railing  bolts  and  the  bolts 
are  fastened  in  by  means  of  grout,  lead  or  sulphur.  To  properly 


238  RETAINING  WALLS 

and  securely  fasten  concrete  railings  to  the  wall  reinforcing  rods 
should  be  incorporated  in  the  coping  while  it  is  being  poured  and 
should  project  a  distance  above  the  top  of  the  coping  to  obtain  a 
good  bond  to  the  hand  rail.  For  all  types  of  railing  provision 
should  be  made  for  the  expansion  due  to  temperature  changes. 

Drainage. — The  presence  of  water  in  a  retained  fill  increases 
the  earth  thrust  in  an  uncertain  but  considerable  amount. 
Again,  to  insure  a  well  founded  roadbed,  water  must  not  be 
permitted  to  accumulate  in  the  fill.  For  these  reasons  means 
are  provided  for  the  removal  of  any  water  that  may  collect  in 
the  fill  behind  the  wall.  The  simplest  method  of  accomplishing 
this  is  to  insert  pipes  in  the  walls  at  frequent  intervals,  permitting 
the  water  to  drain  through  them  and  out  on  the  surrounding 
ground.  To  insure  ample  provision  for  the  run-off  of  the  water 
and  to  prevent  the  pipe  from  silting  up,  a  large  size  pipe,  about 
4  inches  in  diameter  has  proven  to  be  most  satisfactory  as  a 
weep-hole  drain.  The  pipes  should  be  spaced  from  twenty-five  to 
ten-feet  intervals  depending  upon  the  anticipated  conditions  of 
water  accumulation.  That  water  may  be  permitted  to  reach 
these  openings  in  the  wall,  some  rough  drainage  must  be  placed 
at  the  back  of  the  wall.  A  well  planned  wall  will  provide  for  a 
layer  of  broken  stone,  from  6-inches  to  a  foot  in  thickness  upon 
the  back  of  the  wall  and  extending  down  to  the  level  of  the  weep- 
holes.  If  this  method  is  considered  too  expensive,  or  unneces- 
sary for  the  conditions  at  hand,  a  layer  of  broken  stone  may  be 
placed  immediately  around  the  weep-hole,  preventing  the  silt 
from  accumulating  at  the  opening  and  permitting  the  water  to 
drain  off.  Under  no  circumstances  should  the  fill  be  placed 
immediately  against  the  wall  drains. 

It  is  sometimes  objectionable  or  impossible  to  dispose  of  the 
water  through  drains  leading  out  from  the  face  of  the  wall, 
because  of  private  property,  or  important  public  thoroughfares 
adjoining  and  a  regular  sewerage  system  must  be  installed  to 
dispose  of  the  water  through  the  neighboring  sewers.  For 
example  in  the  track  elevation  work  of  the  Rock  Island  Lines.1 

"An  unusual  feature  is  the  provision  of  drainage  wells  in  the  ends  of 
the  retaining  walls  adjacent  to  the  abutments  at  the  subway  bridges. 
These  are  3  feet  by  3  feet  and  extend  to  the  bottom  of  the  wall  (see  Fig. 
128).  There  are  no  weep  holes  through  the  retaining  walls,  but  along 

1  Engineering  News,  Vol.  73,  p.  671. 


ARCHITECTURAL  DETAILS 


239 


d"nfe Drain '- 
to  Catch 
Drain         *• 


4  'Tile  Drain 


6"Tile  Drain 


FIG.  128.— Drainage  of  retained  fill 
carried  to  sewer  system. 


the  backs  of  the  walls  are  laid  inclined  drains  of  6-inch  porous  tile  on  a 
grade  of  0.5  per  cent,  extending  from  subgrade  level  to  6-inch  pipes, 
which  are  imbedded  in  the  rear  part  of  the  walls  and  discharge  to  the 
drainage  wells.  Each  well  has  an  8-inch  connection  to  the  catch-basin 
of  a  city  sewer  as  shown." 

Again  in  the  track  elevation  work  of  the  Philadelphia,  German- 
town  and  Norristown  R.  R.1  The  walls  were  on  private  property 
and  a  layer  of  loose  stone  made  up  in  sizes  varying  from  %  inch 
to  two  feet  were  placed  along  the 
back  of  the  wall.  A  6-inch  vitri- 
fied tile  pipe  was  laid  along  the 
bottom  of  the  wall  below  this 
stone  layer,  on  a  1  per  cent, 
grade,  with  open  joints  and  led 
to  sewers  on  the  cross  streets. 

Another  efficient  method  of 
securing  a  well  drained  fill  is  to 
place  wells  of  broken  stone  at 
each  weep  hole  extending  from 
the  subgrade  of  the  fill  to  the  weep  holes.  In  the  construction 
of  the  retaining  walls  for  the  Hell  Gate  Arch  Approach  (see 
page  127)  it  was  vital  that  no  water  be  allowed  to  accumulate 
in  the  fill  and  wells  were  built  at  each  weep  hole  to  insure  the 
drainage  of  the  earth  work. 

Waterproofing. — The  presence  of  water  in  the  wall  body, 
aside  from  that  left  originally  from  the  concrete  mix,  has  a  harm- 
ful effect  both  on  the  concrete  mass  itself  and  upon  the  face  of 
the  wall.  Generally  it  is  specified  that  some  means  shall  be 
taken  to  keep  the  water  out  of  the  wall.  Retaining  walls  are 
not  made  of  very  rich  mixes  so  that  the  wall  cannot  be  said  to  be 
inherently  water-proof.  It  is  an  easy  matter  to  coat  the  back 
of  the  wall  with  tar  or  asphalt  preparation.  While  it  is  exceed- 
ingly difficult  to  get  an  intact  skin  and  to  keep  it  intact,  care 
exercised  in  placing  the  waterproofing  and  in  preserving  it  from 
accidental  abrasion  after  it  has  been  placed  will  give  a  membrane 
of  sufficient  integrity  to  save  the  face  of  the  wall.  It  is  much 
better  practice  to  place  two  coats  of  waterproofing  upon  the 
back  of  the  wall,  thus  insuring  that  there  are  no  bare  spots  on 
the  wall  back. 


1  Trans.  American  Society  of  Civil  Engineers,  Vol.  Ixxvi,  S.  T.  WAGNER. 


240  RETAINING  WALLS 

Before  placing  the  membrane  of  tar,  it  is  absolutely  necessary 
that  the  wall  be  dry,  free  from  frost  and  well  cleaned.  After 
the  tar  has  been  placed  the  fill  should  be  deposited  with  care  and 
large  boulders  should  not  be  permitted  to  roll  down  and  against 
the  back  of  the  wall.  Where  a  mixed  fill,  rock  and  earth  is  used, 
it  is  good  practice  to  carry  up  the  soft  fill  against  the  back  of  the 
wall  (unless  a  stone  drainage  well  has  been  placed  against  the 
back  of  the  wall)  to  act  as  a  cushion  for  the  rock  fill. 

Where  expansion  joints  occur,  several  layers  of  fabric  coated 
with  hot  tar  are  placed  across  the  joint  to  insure  its  water- 
tightness,  extending  about  a  foot  or  two  on  either  side  of  the 
joint. 

Sub-surface  walls  and  walls  whose  exterior  face  receive  special 
architectural  treatment  to  which  any  moisture  is  damaging, 
must,  of  course,  receive  more  detailed  waterproofing,  involving 
the  extensive  use  of  fabrics,  of  brick  laid  in  an  asphaltic  mastic, 
or  the  possible  additions  of  chemicals  to  the  concrete  mix  itself 
(the  integral  method  of  waterproofing)  all  of  which  fall  without 
the  province  of  the  present  text. 

A  typical  and  well-tried  specification  for  a  tar  coating  for  the 
back  of  the  wall,  may  read  as  follows : 

Coal-tar  shall  be  straight-run  pitch  containing  not  less  than  twenty-five 
percentum  (25%)  and  not  more  than  thirty-two  percentum  (32%)  of  free 
carbon,  and  shall  soften  at  approximately  70°  F.,  and  melt  at  120°  F.,  deter- 
mined by  the  cube  (in  water)  method,  being  a  grade  in  which  distillate  oils 
distilled  therefrom  shall  have  a  specific  gravity  of  1.05. 

Asphalt  shall  consist  of  fluxed  natural  asphalt,  or  asphalt  prepared  by  the 
careful  distillation  of  asphaltic  petroleum  and  shall  comply  with  the  follow- 
ing requirements: 

The  asphalt  shall  contain  in  its  refined  state  not  less  than  ninety-five  per- 
centum (95%)  of  bitumen  soluble  in  cold  carbon  disulphide,  and  at  least 
ninety-eight  and  one-half  percentum  (98.5%)  of  the  bitumen  soluble  in  the 
cold  carbon  disulphide  shall  be  soluble  in  cold  carbon  tetrachloride.  The 
remaining  ingredients  shall  be  such  as  not  to  exert  an  injurious  effect  on  the 
work. 

The  asphalt  shall  not  flash  below  350  degrees  Fahr.,  when  tested  in  the 
New  York  State  Closed  Oil  Tester.  When  twenty  (20)  grams  of  the  mate- 
rial are  heated  for  five  (5)  hours  at  a  temperature  of  325  degrees  F.,  in  a  tin 
box  two  and  one-half  inches  in  diameter  it  shall  lose  not  over  five  percentum 
(5%)  by  weight  nor  shall  the  penetration  at  77  degrees  Fahr.  after  such 
heating  be  less  than  one-half  of  the  original  penetration. 

The  melting  point  of  the  material  shall  be  between  115  degrees  and  135 
degrees  Fahr.,  as  determined  by  the  Kraemer  and  Sarnow  method. 


ARCHITECTURAL  DETAILS  241 

The  consistency  shall  be  determined  by  the  penetration  which  be  between 
75  and  100  at  77°  F. 

A  briquette  of  solid  bitumen  of  cross-section  of  one  square  centimeter 
shall  have  a  ductility  of  not  less  than  twenty  centimeters  at  77°  F.,  the 
material  being  elongated  at  the  rate  of  five  (5)  centimeters  per  minute. 
(Dow  moulds.) 

The  penetrations  indicated*  herein  refer  to  a  depth  of  penetration  in  hun- 
dredth centimeters  of  a  No.  2  cambric  needle  weighted  to  one  hundred 
grams  at  77°  F.,  acting  for  five  seconds. 


if. 


CHAPTER  XI 

LINES    AND    GRADES.      COMPUTATION    OF  VARIOUS  SECTIONS. 
ISOMETRIC  WORKING  SKETCHES.     COST  DATA 

Surveying. — As  an  engineering  structure,  a  retaining  wall 
requires  but  little  more  special  field  work  than  other  masonry 
structures.  The  trenches  within  which  the  wall  is  to  rest  must 
be  staked  out,  the  face  of  the  wall  must  be  laid  out  on  the  con- 
crete bottom  of  the  wall  in  its  correct  location  with  respect  to 
the  property,  or  other  governing  line,  and  finally  the  forms  must 
be  checked  as  to  correct  section  and  location.  As  the  wall  is 
essentially  a  longitudinal  strip,  a  preliminary  line,  parallel  to 
the  face,  or  other  important  line  of  the  wall,  is  staked  out.  This 
forms  the  base  line  of  the  wall  location  work,  and  the  accuracy 
with  which  this  line  is  laid  out  determines  all  the  accuracy  of  the 
lines  subsequently  staked  out  from  this  line.  The  degree  of 
exactness  which  must  be  employed  in  laying  out  the  wall  is 
conditioned  upon  several  factors.  The  presence  of  adjacent 
structures,  the  nearness  of  the  wall  to  important  easement  lines, 
either  public  or  private,  the  necessity  of  tying  other  structures 
to  the  retaining  wall  (or  abutment),  the  proposed  permanence  of 
the  wall,  will  each  control  the  permissible  error  in  the  field-work. 
An  allowable  error  of  one  in  25,000  is  sufficiently  exact  for  any 
type  of  wall,  regardless  of  the  degree  of  exactness  required  and 
larger  error  factors  should  be  used  for  less  important  structures. 

The  importance  of  the  base  line  with  reference  to  the  field 
work  which  follows  and  is  dependent  upon  it  makes  it  necessary 
that  it  be  laid  out  at  a  distance  away  from  the  work  that  will 
keep  it  safely  out  of  the  construction  way,  and  yet  close  enough 
that  it  can  readily  be  employed  as  a  reference  line.  If  th  loca- 
tion of  the  work  permits  the  line  should  be  about  25  feet  away 
from  the  wall  line  and  referenced  at  frequent  intervals  to  fixed 
land  marks.  It  should  be  tied  in  to  other  important  lines  of 
permanent  nature,  such  as  city  monument?  lines,  the  main  rail- 
road survey  lines  and  such  lines  as  control  more  or  less,  the  loca- 
tion of  the  easement  lines  of  the  wall. 

242 


LINES  AND  GRADES  243 

In  conjunction  with  the  location  of  the  base  line,  a  run  of 
benches  is  made,  safely  established,  so  that  the  progress  of  con- 
struction will  not  disturb  them.  The  accuracy  of  this  run  need 
not  be  high,  unless  steel  structures  are  to  be  tied  into  the  wall 
(e.g.  abutments  supporting  steel  bridges;  retaining  walls  carry- 
ing building  walls  upon  them,  etc.). 

It  is  patent,  that  in  the  establishment  of  both  the  base  line  and 
the  bench  run,  points  must  be  selected  that  can  readily  be  found 
and  used  for  the  construction  work.  This  is  a  matter  of  judg- 
ment, tempered  by  much  field  experience  and  vexatious  delays 
must  occur  through  poor  selection  of  important  surveying  points. 

Construction  Lines. — The  base  line  as  above  described  is  not 
used  directly  to  stake  out  the  construction  work.  It  is  cus- 
tomary to  place  a  line  about  five  feet  from  the  face  of  the  wall, 
and  where  possible,  another  line  ten  feet  from  the  face,  and 
both  parallel  to  the  base  line,  which  lines  are  directly  employed 
by  the  mechanics  to  lay  out  the  excavation  lines  and  the 
concrete  lines.  As  the  lines  are  destroyed  in  the  ordinary  course 
of  construction,  they  may  easily  be  restored,  where  necessary, 
by  recourse  to  the  permanent  base  line.  On  tangent  walls,  net 
line  stakes  (i.e.  the  actual  wall  lines)  may  be  placed  at  twenty 
to  twenty-five  foot  intervals.  On  curves,  they  should  be  placed 
close  enough,  that  the  chords  do  not  diverge  more  than  the  per- 
missible limit  from  the  true  arc 
of  the  wall.  For  the  excavation  i  ,- Amoun 

lines,  rough  work  is,  of  course 
permissible.  For  the  concrete 
lines  more  refinement  is  re- 

.      _„      ,    .          .        . ,  FIG.  129. — Length  of  chord  for  per- 

quired.   To  determine  the  proper          missible  amount  of  flattening. 
chord    length,  L,  to  be  used  in 

staking  out  the  wall,  so  that  its  middle  ordinate  will  not  exceed 
the  permissible  allowance  a  (see  Fig.  129),  note  that  from  the 
approximate  parabolic  relation  that  the  offset  y  to  an  arc,  at  a 
distance  x  from  the  point  of  tangency  is  given  by  the  formula 

y  =  x*/2R 

where  R  is  the  radius  of  the  arc.  Employing  this  formula  in  the 
present  case 

a  =  L2/SR 
It  is  generally  specified  that  the  flatness  of  the  wall  shall  not 


244 


RETAINING  WALLS 


exceed  J£  of  an  inch,  or  0.01  feet.     This  last  equation  when  solved 
for  L,  using  the  value  0.01  for  a  is  then 


To  aid  in  the  use  of  this  equation,  Table  No.  39  is  given  here- 
with showing  the  necessary  chord  length  to  be  used  for  any 
assigned  radius  of  arc,  that  the  chord  offset  shall  not  exceed 
one-eighth  of  an  inch.  For  example,  by  reference  to  the  table, 
a  radius  of  800  feet  makes  it  necessary  to  stake  out  the  wall  in 
eight-foot  chord  lengths,  while  a  radius  of  8,000  feet  permits  the 
use  of  twenty-five-foot  chords. 

TABLE  39.  —  MAXIMUM  CHORD  LENGTHS 


R 

L 

R 

L 

R 

L 

R 

L 

R 

L 

50 

2.0 

325 

5.1 

700 

7.5 

1250 

10.0 

4500 

19.0 

75 

2.5 

350 

5.3X 

750 

7.8 

1300 

10.2 

5000 

20.0 

100 

2.8 

375 

5.5 

800 

8.0 

1350 

10.4 

5500 

21.0 

125 

3.2 

400 

5.7 

850 

8.2 

1400 

10.6 

6000 

21.9 

150 

3.5 

425 

5.8 

900 

8.5 

1450 

10.8 

6500 

22.8 

175 

3.7 

450 

6.0 

950 

8.7 

1500 

11.0 

7000 

23.7 

200 

4.0 

475 

6.2 

1000 

9.0 

2000 

12.7 

7500 

24.5 

225 

4.2 

500 

6.3 

1050 

9.2 

2500 

14.2 

8000 

25.0 

250 

4.5 

550 

6.6 

1100 

9.4 

3000 

15.5 

275 

4.7 

600 

6.9 

1150 

9.6 

3500 

16.7 

300 

4.9 

650 

7.2 

1200 

9.8 

4000 

17.9 

The  bottom  of  the  wall,  whether  of  concrete  or  other  masonry 
should  not  necessarily  fill  the  trench  unless  this  has  been  trimmed 
with  unusual  care.  If  the  net-line  stakes  have  been  lost  in  the 
excavation,  these  should  be  restored,  and  the  proper  bottom  lines 
given  for  the  masonry  footing.  For  grades,  stakes  may  be  driven 
into  the  side  of  the  cut  at  the  required  elevation,  or  at  a  stated 
distance  above  this  line.  For  the  elevation  of  the  bottom  of  the 
wall  the  same  stakes  may  be  employed. 

Forms. — With  the  bottom  concrete  in,  it  is  necessary  to  give 
some  line  to  commence  the  form  work.  A  very  serviceable 
method  is  that  of  nailing  a  molding  strip  to  the  concrete  bottom, 
marking  the  inside  of  the  lagging  of  the  form  (see  Fig.  130). 
With  this  line  in  place  the  face  forms  may  be  set  and  the  rear 
forms  placed  at  the  required  distance  away  as  specified  on  the 


LINES  AND  GRADES  245 

plans.  After  the  forms  are  assembled,  wired  and  braced,  they 
may  be  rechecked  from  the  reference  line,  and  then  plumbed  to 
see  that  the  section  meets  that  theoretically  required.  The 
proper  grades  at  which  to  make  the  breaks  in  the  wall  section,  if 
there  be  any,  and  the  grade  for  the  top  of  the  wall,  are  most 
commonly  given  by  nails  driven  in  the  side  of  the  form  at  these 
elevations. 

Concrete  Forms 


Concrete  Net       :-  Strip  of  Molding 
Lines  Nailed  to  Footing 


FIG.  130.  —  Method  of  lining  in  concrete  forms. 

Computation  of  Volumes.  —  When  the  section  of  a  retaining 
wall  remains  constant  between  two  given  points,  its  volume  is  the 
product  of  the  area  of  the  section  by  the  distance  between  the  two 
points.  Generally  the  section  of  the  wall  varies,  the  top  of  the 
wall  following  a  given  grade.  Breaks  in  the  width  of  the  wall,  or 
in  other  but  the  vertical  dimensions,  are  made  at  the  expansion 
joints,  so  that  between  two  adjacent  expansion  joints  the  width 
of  wall  at  the  coping  and  at  the  base  remain  constant.  The 
volume  of  a  wall,  whose  coping  and  base  widths  are  respectively 
a  and  6,  and  whose  heights  at  the  beginning  and  end  of  the  section 
are  hi  and  7i2,  respectively,  is 

V  =      (a  +  6)(Ai  +  hz) 


To  get  the  volume  of  sections  of  the  wall  which  are  irregular 
because  of  breaks  in  the  wall,  or  because  of  intersections  with 
other  walls,  it  is  essential  that  a  careful  and  detailed  drawing  be 
made.  It  is  difficult  to  show  clearly  the  volume  in  question 
when  the  drawing  merely  gives  a  two-dimension  section.  For  this 
reason  isometric  drawings  may  serve  to  bring  out  clearly  and 
exactly  all  the  dimension  necessary  to  obtain  the  volume  of  the 
portion  sought.  To  make  the  isometric  drawing  correct  to  scale 
and  to  be  able  to  interpret  mathematically  the  lengths  scaled 
from  the  isometric  drawing  the  following  matter  gives  some 
formulas  and  tables  which  should  serve  to  make  the  isometric 
layout  as  easy  to  handle  as  the  plane  detail  drawing.1 

1  See  Engineering  News-Record,  April  3,  1919,  p.  661. 


246  RETAINING  WALLS 

It  is  assumed  that  the  isometric  taxes  of  the  figure  have  so  been 
chosen  that  all  the  important  lines  of  the  figure  lie  in  planes 
parallel  to  the  axes.  The  following  theorems  apply  solely  to 
such  lines.  Lines  parallel  to  the  axes  are  shown  correctly  to 
length  by  the  principles  of  isometric  projection.  Lines  not 
parallel  are  not  shown  correctly  to  length.  To  obtain  the  angles 
which  these  lines  make  in  actual  space,  and  the  actual  lengths 
of  such  lines  and  conversely,  the  lengths  of  such  lines  in  isometric 
projection  and  the  angles  which  they  make  with  the  isometric 
axes,  refer  to  Figs.  131  and  132.  In  Fig.  131,  thelineL  has  pro- 


c  c  c 

FIG.  131.  FIG.  132. 

The  plane  and  isometric  triangles. 

jections  b  and  c  and  makes  an  angle  <£  with  the  projection  c. 

In  isometric  projection  the  length  L  becomes  either  Li  or  L,- 

depending  upon  whether  it  subtends  an  angle  of  120°  or  60°. 

The  angle  <£  is  again  <£;  or  #,-  in  isometric  projection.     The  lengths 

b  and  c  remain  unchanged,     tan  <f>  =  b/c.     Referring  to  Figure 

132,  by  the  law  of  sines 

b/c  =  sin  0f/sin  (180°  -  120°  -  <£<) 
b/c  =  sin  4>,-/sm  (180°  -    '  60°  -  <fc) 

From  which  two  equations, 


2  cot  0- 


Table  40  gives  the  values  of  <£»  and  <£/  for  the  several  values  of  </>. 
Referring  to  Figs.  131,  132. 
L2   =  b2  +  c2 

LS  =  b2  +  c2  -  2bc  cos  120°  =  62  +  c2  +  be 
L?  =  =  b2  +  c2  -  be 

b  =  L  cos  0  and       c  =  L  sin  <£ 

Substituting  these  values  in  the  preceding  equations  there  is 
finally 

Li  =  kL;     Lj  =  jL 
where  k2  =  1  +  sin  0  cos  $;    j2  —  1  —  sin  4>  cos  0. 


LINES  AND  GRADES 


247 


Table  40  gives  a  series  of  values  of  k  and  j  for  the  run  of 
values  of  0. 

TABLE  40. — ISOMETRIC  FUNCTIONS 


0 

0» 

<t>i 

k 

j 

0° 

0° 

0° 

1.00 

1.00 

5 

4 

5 

1.04 

0.95 

10 

8 

10 

1.09 

0.90 

15 

12 

15 

1.12 

0.87 

20 

15 

21 

1.15 

0.82 

25 

18 

28 

1.18 

0.79 

30 

21 

35 

1.20 

0.75 

35 

24 

43 

1.21 

0.73 

40 

•  27 

51 

1.22 

0.71 

45 

30   . 

60 

1.23 

0.71 

Fig.  133  gives  an  illustration  of  some  wall  details  shown  iso- 
rnetrically  and  properly  scaled  and  dimensioned  (all  dimensions 


For  Lines  Parallel  toAF 
true  Dimensions  are  ecjua/ 
to  Scaled  Dimensions 
Divided  by  k=i:il 


Estimate  Volume  of  Large  Section  to  Plane  DHGJ 
Estimate  Volume  of  Small  Section  to  Plane  CABD 
Estimate  Volume  of  Irregular  Junction  as  Follows : 
Volume  of  Right  Prism  -A  EGK- Altitude  DE 
Less,  Volume  of  Right  Pyramid-Base  BMHE 
Altitude  ED 


FIG.  133. — The   isometric   detail    and  its   application   to   the   computation  of 

volumes. 


248  RETAINING  WALLS 

shown  are  the  true  ones,  the  isometric  lengths  as  shown  having 
been  corrected  by  means  of  the  tables  above. 

Cost  Data. — The  compilation  of  worth-while  cost  data  is 
conditioned  upon  the  proper  valuation  of  the  relative  operations 
involved  in  the  piece  of  work  under  analysis  as  well  as  a  correct 
understanding  as  to  how  much  of  the  work  is  standard  in  con- 
nection with  retaining  wall  construction  and  how  much  is  peculiar 
to  the  individual  piece  of  work  in  question.  Merely  gathering 
cost  statistics  without  an  intelligent  interpretation  of  the  opera- 
tions affecting  or  controlling  costs  is  a  valueless  and  time  wasting 
procedure. 

Cost  analysis  in  general  may  be  said  to  serve  two  purposes. 
It  furnishes  an  accounting  of  work  already  done,  in  order  that 
proper  disbursements  may  finally  be  made  and  a  correct  financial 
history  compiled  of  the  job  in  question.  In  this  sense  it  is 
properly  an  accounting  job,  based  upon  payroll  and  material 
forms  prepared  by  the  timekeeper.  It  may  also  be  an  antici- 
patory analysis  of  work  to  be  done  and  then  comes  within 
the  province  of  an  engineer  preparing  such  an  estimate.  Proper 
attention  to  the  former  purpose  of  cost  data  is  of  course  essential 
that  the  latter  purpose  may  be  efficiently  carried  out  and  the 
more  voluminous  the  files  of  cost  accounts  (intelligently  kept) 
the  better  able  is  the  engineer  to  make  a  scientific  prediction  of 
the  cost  of  future  work. 

That  a  true  comparison  may  be  made  of  the  relative  value 
of  the  various  types  of  retaining  walls,  it  is  apparent  that  the 
elements  entering  into  the  cost  data  must  be  properly  weighted,  so 
that  items  of  cost  unique  to  a  peculiar  environment  be  disregarded. 
For  this  purpose,  it  is  best  that  cost  data  be  reduced,  as  far  as  is 
practicable,  to  fundamental  and  elemental  operations,  independ- 
ent, more  or  less,  of  the  peculiar  character  of  any  piece  of 
construction. 

Cost  may  be  divided  into  several  general  subdivisions :  Labor 
cost;  material  cost;  plant  cost  and  general  administrative 
expenses.  The  first  item,  the  labor  cost,  is  the  uncertain  item, 
and  one  requiring  experience  and  judgment  in  its  proper  deter- 
mination. Material  costs  are  simple,  are  easily  compiled;  can 
easily  be  anticipated  and  with  a  proper  allowance  for  the  wastage 
involved  in  the  several  operation  are  estimated  with  a  high 
degree  of  accuracy.  Plant  cost,  while  possibly  not  so  easily 
compiled  or  anticipated  as  material  cost,  should  not,  at  least 


LINES  AND  GRADES  249 

to  the  engineer  with  a  moderate  amount  of  experience,  prove 
difficult  of  computation.  In  a  previous  chapter  the  character 
and  the  distribution  of  plant  employed  for  a  number  of  pieces  of 
typical  retaining  wall  construction  may  furnish  a  good  working 
clue  to  the  type  most  suited  to  the  work  under  analysis.  General 
administrative  expenses  will  cover  office  expenses,  salaries  of  the 
executives,  insurance  upon  the  labor,  miscellaneous  casualty  and 
public  liability  insurance,  minor  expenses  in  connection  with  the 
prosecution  of  the  work,  such  as  telephone,  fares,  taxes,  etc. 
This  item  is  usually  termed  the  overhead  of  the  work  and  is 
spread  over  all  the  items  entering  into  the  construction  of  a  wall. 
While  of  an  indefinite  character,  it  must  be  properly  ascertained 
or  anticipated  in  order  to  be  included  in  the  estimated  cost. 
It  must  be  remembered  that  it  is  a  constant  charge  carried 
continuously,  regardless  of  the  weather  or  other  delays  and  in 
work  of  long  duration,  may  effect  materially  the  cost  of  the  opera- 
tions. Blanket  percentages  added  to  cover  items  of  this  nature, 
while  excusable  in  small  work,  are  apt  to  work  hardships  upon 
large  work  unless  the  percentage  factor  so  applied  is  the  result 
of  data  compiled  from  several  jobs  of  similar  nature.  Naturally 
the  number  of  items  of  uncertain  amount  appearing  in  an  esti- 
mate of  future  work  will  be  in  inverse  proportion  to  the  amount 
of  experience  of  the  engineer  preparing  such  estimates. 

Labor  Costs. — Without  entering  into  a  detailed  analysis  of 
the  various  labor  elements  involved  in  wall  construction,1  some 
general  labor  costs  may  be  presented  to  guide  an  estimator  in 
preparing  a  bid  for  contemplated  work.  Before  employing 
such  data  it  is  well  to  read  again  (chapter  on  "  Plant ")  the  impor- 
tant bearing  of  plant  selection  and  arrangement  upon  the  cost 
of  labor.  A  good  bid  is  not  one  that  contains  merely  a  carefully 
and  detailed  analysis  of  the  cost  of  the  labor.  It  must  plan  a 
scheme  of  the  work  together  with  the  amount  of  plant  to  be  had 
and  the  character  of  the  labor  to  operate  it.  Haphazard  bidding 
or  snap  judgment  estimates  are  unpardonable  in  all  but  the  most 
experienced  of  contractors  and  engineers,  and  must  eventually 
lead  to  financial  disaster.  Such  figures  and  quoted  estimates  of 
the  cost  of  work  as  are  given  below  must  be  used  in  light  of  the 
above  remarks. 

The  material  for  the  wall' is  taken  from  the  point  of  delivery 

1  See  DANA,  "Cost  Data,"  GILLETTE  "Handbook  of  Cost  Data;"  TAYLOR 
and  THOMSON,  "Concrete  Costs." 

UNIVERSITY   OF   CALIFORNIA 
DEPARTMENT  OF  CIVIL   ENGINEER 


250  RETAINING  WALLS 

and  brought  to  the  site  of  the  work  either  at  a  contracted  price 
per  yard  (which  price  may  be  ascertained  at  the  time  of  preparing 
the  bid)  or  if  delivered  F.O.B.  nearest  railroad  station  or 
lighterage  dock  may  be  hauled  by  hired  team  or  auto  truck. 
With  the  latter  method,  the  length  of  haul  will  determine  the 
average  number  of  trips  that  the  trucks  can  make,  and  knowing 
the  load  that  can  be  carried,  the  price  per  yard  for  delivering 
the  material  can  be  computed  with  no  great  difficulty.  An 
analysis  of  the  cost  of  several  pieces  of  work,  follows.  The 
files  of  the  Engineering  Press  may  be  used  to  examine  the  cost 
of  numerous  pieces  of  work. 

From  TAYLOR  and  THOMPSON  " Concrete  Costs,"  p.  16: 

Cantilever  watt,  16  feet  high,  250  feet  long;  common  labor  $2.00  per  day, 
carpenters  $3.82  per  day.  Concrete  yardage  277  cubic  yards. 

Cost  of  labor  of  forms  per  cubic  yard  of  concrete.  ...  $2.75 

Total  cost  of  forms  per  cubic  yard  of  concrete $3.91 

Cost  of  material  per  cubic  yard  of  concrete $3 . 57 

Cost  of  mixing  and  placing  concrete  per  cubic  yard .  .  $1 . 35 
Total  cost  of  concrete  in  place   (including  superin- 
tendence)   $12.0.3 

Cantilever  wall  16  feet  high.  Labor  20  cents  per  hour;  carpenters  50  cents 
per  hour. 

Total  cost  of  forms  per  cubic  yard  of  concrete $3 . 60 

Cost  of  concrete  material  per  cubic  yard 4 .75 

Cost  of  mixing  and  placing  the  concrete 1 . 25 

Cantilever  wall  8  feet  high.  Labor  20  cents  per  hour;  carpenters  50  cents 
per  hour. 

Total  cost  of  forms  per  cubic  yard  of  concrete $6 . 23 

Total  cost  of  material  per  cubic  yard  of  concrete.  ...       4.75 
Cost  of  mixing  and  placing  per  cubic  yard 1 . 25 

A  resume  of  the  total  labor  cost  of  pouring  retaining  walls  of 
both  gravity  and  reinforced  "L"  type,  averaging  about  35  feet 
in  height  is  as  follows: 

Gravity  Type  1935,  cubic  yards  of  concrete.  Plant  used  was 
two  small  batch  mixers,  the  concrete  wheeled  to  the  forms  and 
poured  in.1 

1  See  " Enlarging  an  Old  Retaining  Wall,"  for  a  detailed  description  of  the 
methods  and  plantnised,  Engineering  News,  Sept.  8,  1915. 


LINES  AND  GRADES  251 

The  forms  were  used  on  the  average  about  four  times. 

Foreman,  175  days  at  $5.00  per  day $875.00 

Carpenters,  190  days  at  $3.50  per  day $665 . 00 

Engineer,  46  days  at  $5.00  per  day 230 . 00 

Laborers,  926  days  at  $1.75  per  day 1620 . 50 

Teams,  21  days  at  $3.50  per  day 73 . 50 

Timbermen,  20  days  at  $3.00  per  day 60 . 00 

Masons,  37  days  at  $4.00  per  day 148 . 00 

Riggers,  14  days  at  $3.00  per  day 42 . 00 

Watchmen,  33  days  at  $1.00  per  day 33 . 00 

Total  labor  cost $3747.00 

This  makes  the  labor  cost  per  yard,  exclusive  of  all  overhead 
insurance,  plants  charges  etc.,  $1.94  per  cubic  yard  of  concrete. 

A  similar  detailed  labor  cost  to  pour  a  "L"  shaped  cantilever  wall,  in- 
volving a  yardage  of  1697  cubic  yards  is: 

Foremen,  197  days  at  $5.00  per  day $985 . 00 

Carpenters,  503  days  at  $3.50  per  day 1760.50 

Engineer,  37  days  at  $5.00  per  day 185 . 00 

Riggers,  24  days  at  $3.00  per  day 72 . 00 

Laborers,  1197  days  at  $1.75  per  day 2094.75 

Masons,  55  days  at  $4.00  per  day 220.00 

Teams,  51  days  at  $3.50  per  day 178 . 50 

Watchmen,  124  days  at  $1.00  per  day 124.00 

Total  labor  cost $5619 . 75 

The  unit  labor  cost  per  cubic  yard  for  pouring  this  type  of 
wall,  exclusive  of  all  overhead  charges  as  above  enumerated  is 
$3.31  per  cubic  yard. 

While  endless  data  might  be  furnished  of  the  cost  of  existing 
work,  conditions  are  usually  too  unique  to  make  such  data  of 
general  usefulness.  Unit  costs  as  quoted  above  may  fill  in 
uncertain  data  in  a  bid,  when  properly  altered  to  take  care  of 
changed  labor  rates.  The  labor  cost  on  a  retaining  wall,  roughly, 
averages  about  one-quarter  the  total  cost  of  the  wall.  Barring 
unforseen  contingencies  an  estimator  with  a  fair  knowledge  of 
construction  work  should  be  able  to  anticipate  the  labor  cost 
within  20  per  cent,  of  its  correct  final  value.  Should  the  dis- 
crepancy amount  to  the  limiting  value  of  20  per  cent.,  in  the 
final  data  it  will  amount  to  merely  5  per  cent,  of  the  total  cost 
of  the  work.  Estimates  of  work  can  hardly  be  expected  to 
reach  a  higher  degree  of  accuracy  than  this. 

As  an  example  of  the  analysis  of  a  proposed  piece  of  work,  let 


252  RETAINING  WALLS 

it  be  required  to  determine  the  cost  of  constructing  a  retaining 
wall  about  1,000  feet  long,  40  feet  high,  with  a  yardage  of  about 
10,000  cubic  yards.  One  year  is  the  allotted  time  in  which  to 
construct  the  wall.  The  wall  is  a  cantilever  type. 

Plant. — A  mixer  of  about  100  yards  per  day  capacity  (a  J^  to 
%  yard  batch  mixer  will  easily  satisfy  this  requirement)  should 
pour  the  required  yardage  of  concrete  with  an  ample  time  margin. 
This  mixer  should  be  obtained  in  the  neighborhood  of  about 
$1,000.  The  other  plant  requirements,  such  as  wheelbarrows, 
shovels,  etc.;  shanties  for  storing  cement  and  tools,  for  temporary 
offices;  lumber  for  runways  for  pouring  the  concrete  etc.,  should 
not  cost  more  than  an  additional  $1,000  making  the  total  plant 
charge  $2,000. 

Materials. — Assuming  that  the  wall  is  a  1:2.5:5  mixture  of 
concrete,  there  will  be  required  about  1.2  barrels  of  cement  for 
each  yard  of  concrete  placed.  Theoretically  about  10,000  yards 
of  stone  and  5,000  yards  of  sand  will  be  required.  To  allow  for 
wastage  of  all  kinds  these  quantities  will  be  increased  10  per 
cent.  It  will  be  assumed  that  the  materials  will  be  delivered 
on  the  job,  where  required  for  the  following  unit  prices;  cement 
$3.50  per  barrel  (net,  no  allowance  for  bags);  stone  for  $2.50 
per  yard  and  sand  for  $2.00  per  yard.  The  material  totals  are 
then 

13,200  bbls.  cement  at  $3.50 $46,200 

11,000  yards  stone  at  $2.50 27,500 

5,500  yards  sand  at  $2.00. 11,000 

The  total  material  will  cost $84,700 

Form  Lumber. — Assume  that  2  inch  tongue  and  grooved 
sheeting  will  be  used  to  make  the  form  panels.  Allow  about 
20  per  cent,  wastage  of  forms  each  time  the  forms  are  stripped 
(this  is  equivalent  to  a  form  use  of  five  times).  The  area  of 
wall  surface  that  must  be  covered  with  new  form  lumber  is  then 
(allowing  a  footing  thickness  of  four  feet) 

36  X  2 X  10°°  =  14,400  square  feet. 
o 

To  allow  for  the  joists,  rangers,  bracing  etc.,  and  to  allow  for  was- 
tage in  material  due  to  cutting  it  to  required  lengths,  it  is  cus- 
tomary to  double  the  board  feet  required  for  the  sheeting. 
(Exactly,  the  forms  may  be  designed  as  outlined  in  the  chapter 


LINES  AND  GRADES  253 

on  FORMS,  and  detailed  as  shown  in  the  problem  accompanying 
the  chapter,  and  the  required  amount  of  timber  taken  from  these 
estimates.  An  estimate  of  the  cost  of  the  work,  does  not,  how- 
ever, justify  such  refinement,  and  it  is  better  to  use  the  rule  of 
thumb  method  just  stated.)  Since  the  sheeting  is  to  be  2 
inches  thick,  the  total  lumber  requirements  are  4  board  feet  for 
every  square  foot  of  new  lumber  surface.-  With  a  price  of  $75 
per  M  for  timber  delivered  on  the  job,  tongue  and  grooved,  the 
timber  cost  is 

14.4  X  4     at     $75  =  $4,320 

Labor  Costs. — To  get  the  total  labor  costs  on  the  wall,  the 
analysis  of  the  cost  of  the  reinforced  concrete  wall  at  last  outlined 
may  be  used  with  the  following  revised  rates  of  labor:  Foreman 
$8.00  per  day,  Engineer  and  Carpenters,  $7.00  per  day;  laborers 
$4.00  per  day  and  the  other  items  in  keeping.  This  will  pracr 
tically  double  the  unit  cost  of  labor  as  given.  The  unit  cost  is 
then  about  $6.75  per  yard,  or  the  total  cost  is  $67,500.  To  this 
must  be  added  the  item  of  insurance,  amounting  to  about  10  per 
cent,  of  the  labor  total,  or  $6,750. 

Overhead. — The  work  will  require  the  employment  of  a  super- 
intendent for  one  year  ($4,000)  and  a  timekeeper  ($1,500) 
Miscellaneous  expenses  around  the  work  should  not  exceed 
$1,000,  making  the  field  overhead  about  $6,500. 

The  office  overhead  is  indeterminate,  depending  upon  the 
number  of  jobs  going  on  at  one  time.  This  factor  will  be  omitted 
here. 

The  rods  are  usually  quoted  at  a  separate  unit  price  and  are 
not  mentioned  here. 

To  summarize: 

Plant $2,000 

Materials 84,700 

Lumber 4,320 

Labor  (and  Ins.) 74,250 

Overhead 6,500 

Total $171,770 

With  an  allowance  for  profit  the  wall  will  be  estimated  in  the 
neighborhood  of  $200,000,  or  at  a  unit  cost  of  $20.00  per  cubic 
yard. 


SPECIFICATIONS 

General  Layout  of  Work. — The  retaining  walls  to  be  constructed  under 
this  contract  are  shown  on  Plans  Nos.  to  inclusive.  These  specifica- 
tions and  the  plans  are  intended  to  be  consistent  and  where  any  apparent 
inconsistency  appears  the  interpretation  shall  convey  the  intent  of  the  best 
work  and  construction. 

Classes  of  Work. — The  retaining  walls  shall  be  classified  for  payment  as 
follows : 

Class  A. — Walls  without  reinforcement,  marked  A  on  the  plans,  of  what- 
ever height  indicated.  I 

Class  B. — Reinforced  concrete  walls  up  to  but  not  including  twenty  (20) 
feet  in  height  from  subgrade  to  top  of  coping. 

Class  C. — Reinforced  concrete  walls  from  twenty  (20)  feet  up  to  but  not 
including  thirty  (30)  feet  from  subgrade  to  top  of  coping. 

Class  D. — Reinforced  concrete  walls  over  thirty  (30  feet)  in  height  from 
subgrade  to  top  of  coping. 

Class  E. — Walls  of  cement  rubble  masonry  of  whatever  height  indicated. 
Payment. — Payment  for  the  walls  as  indicated  shall  include  the  furnish- 
ing of  all  labor  and  materials  necessary,  including  the  cost  of  all  scaffolding, 
forms  and  the  cost  ol  removing  the  same;  also  the  cost  of  finishing  the  face 
of  the  wall  where  a  rubbed  finish  is  indicated. 

Concrete  Proportions. — Concrete  for  class  A  walls  shall  be  mixed  in  the 
proportions  of  one  part  cement,  two  and  one-half  parts  of  sand  and  five 
parts  of  stone  or  gravel,  by  volume. 

Concrete  for  reinforced  concrete  walls  (classes  B,  C  and  D)  shall  be  mixed 
in  the  proportions  of  one  part  cement,  two  parts  sand  and  four  parts  of 
stone  or  gravel,  by  volume. 

Cement. — The  cement  shall  be  Portland  Cement  of  a  brand  that  has  been 
on  the  market  for  the  last  five  years. 

(Insert  here  the  details  of  the  properties  of  cement  as  has  been  given 
on  pages  214  to  215.) 

Sand. — Sand  for  use  in  making  the  concrete  shall  be  clean  and  well 
graded,  not  exceeding  %  inch  in  size.  Not  more  than  six  per  centum  (6%) 
by  weight  shall  pass  a  100  mesh  screen.  It  shall  contain  not  more  than  three 
per  centum  (3%)  by  weight  of  foreign  matter. 

Broken  Stone. — Stone  for  concrete  shall  be  a  clean  sound,  hard  broken 
limestone  or  trap  rock  and  graded  from  three-eighths  (%)  of  an  inch  in  di- 
ameter up  to  one  and  three-quarters  (1%)  inches  in  diameter.  Where  the 
thickness  of  the  concrete  wall  is  twelve  inches  or  less  in  thickness  the  size 
of  the  stone  shall  not  exceed  three-quarters  (%)  of  an  inch  in  diameter. 
It  shall  be  screened  and  washed  to  remove  all  impurities  and  shall  be  care- 
fully stored  along  the  site  of  the  work  to  prevent  the  gathering  of  any  foreign 
matter  in  it. 

254 


SPECIFICATIONS  255 

Gravel. — Gravel  shall  be  screened,  cleaned  and  graded  in  the  same  man- 
ner as  the  broken  stone. 

Use  of  Large  Stone. — In  Class  A  walls  (and  in  these  walls  only)  where 
the  thickness  of  the  wall  exceeds  thirty  (30)  inches  the  contracter  will  be 
permitted  to  imbed  stones  of  at  least  12  inches  in  thickness  not  closer  than 
four  (4)  inches  to  the  face  of  the  form  and  not  closer  than  six  (6)  inches  to 
each  other.  The  stones  shall  be  sound,  clean  stones  and  shall  be  carefully 
placed  in  the  concrete. 

Concrete.— Concrete  shall  be  mixed  by  machine.  In  case  of  emergency 
it  shall  be  within  the  discretion  of  the  Engineer  to  state  whether  the  mixing 
shall  proceed  by  hand. 

It  is  the  very  essence  of  these  specifications  that  the  water  content  of 
the  concrete  mix  by  kept  low.  No  machine  mixer  shall  be  used  that  is  not 
equipped  with  a  tank  or  other  device  for  supplying  a  measured  amount  of 
water  to  each  batch  of  concrete  and  a  competent  operator  shall  be  in  atten- 
dance upon  the  machine. 

The  Engineer,  or  his  duly  authorized  representative  shall  decide  upon 
the  length  of  time  each  batch  shall  be  mixed  and  upon  the  amount  of  water 
that  shall  go  into  each  batch. 

The  contractor  shall  permit  the  Engineer  to  take  samples  of  the  concrete 
mix  to  be  tested  and  no  charges  shall  be  made  for  material  taken  for  such 
purposes . 

The  use  of  a  continuous  mixer  is  forbidden  and  a  mixer  that  is  found 
incapable  of  delivering  a  concrete  in  conformity  with  the  specifications  shall 
be  removed  from  the  work  and  a  mixer  substituted  for  it  that  is  capable  of 
mixing  concrete  in  accordance  with  these  specifications. 

Concrete  shall  be  conveyed  to  the  forms  in  water-tight  conveyances  and 
shall  be  dropped  vertically  into  the  forms..  It  shall  then  be  shovelled  into 
place  and  thoroughly  compacted  and  rammed  to  insure  a  concrete  of  uniform 
density. 

Spades  or  other  special  tools  shall  be  used  on  the  concrete  to  insure  a  free 
circulation  of  the  grout  around  the  reinforcing  bars  and  against  the  face  of 
the  forms. 

Forms. — The  forms  for  concrete  shall  be  made  of  stout  tongue  and  grooved 
sheeting,  properly  supported  and  braced  and  of  strength  sufficient  to  meet 
the  concrete  pressures.  If  so  required  the  contractor  shall  submit  to  the 
engineer  plans  of  the  form  work  and  bracing. 

Before  pouring  the  forms  shall  be  oiled,  or  thoroughly  wetted  and  before 
reusing  shall  be  cleaned  of  all  adhering  cement,  dirt,  etc.,  to  insure  a  smooth 
face  on  all  exposed  concrete  work. 

The  joints  shall  be  water-tight  and  shall  be  carefully  inspected  while 
the  pouring  is  in  progress  to  prevent  the  escape  of  any  grout. 

Concrete  shall  set  at  least  twenty-four  (24)  hours  before  the  tie-rods  are 
loosened  or  any  of  the  sheeting  removed.  This  time  shall  be  increased  when 
the  temperature  of  the  air  drops  below  sixty  (60)  degrees  Fahr.  Forms  shall 
be  stripped  in  the  presence  of  the  Engineer,  if  the  contractor  is  so  directed. 

Placing  Fill. — No  fill  shall  be  deposited  behind  the  walls  until  ten  days 
have  elapsed  since  the  walls  were  poured  and  not  until  the  assent  of  the 
Engineer  or  his  duly  authorized  representative  has  been  obtained. 


256  RETAINING  WALLS 

Defective  Work. — If  upon  stripping  the  forms  there  is  evidence  of  any 
defective  work,  such  defective  work  shall  immediately  be  repaired  and  the 
surface  of  the  wall  finished  in  a  manner  that  will  present  as  little  evidence 
of  such  defective  work  as  possible. 

Evidence  of  extensive  defective  work  shall  be  sufficient  cause  to  order  the 
contractor  to  remove  portions  of  the  work  showing  such  defective  work  and 
all  such  repairs  and  reconstruction  work  shall  be  made  at  the  contractor's 
own  expense. 

Concrete  Work  in  Winter  Weather. — When  the  temperature  of  the  air 
drops  below  45  degrees  Fahr.  it  shall  be  within  the  discretion  of  the  Engineer 
to  order  the  contractor  to  heat  the  concrete  materials  before  pouring  them 
into  the  forms. 

No  concrete  shall  be  deposited  in  the  forms  in  freezing  temperature  that 
has  not  been  mixed  with  materials  heated  by  means  of  suitable  appliances 
so  that  the  temperature  of  the  concrete  upon  being  placed  in  the  forms  shall 
not  be  less  than  60  degrees  F.  Concrete  deposited  in  freezing  weather  shall 
be  protected  while  setting  by  means  of  salt  hay,  tarpaulin,  canvas,  or  by 
other  devices  which  will  maintain  the  temperature  of  the  concrete  above 
freezing  until  it  has  set. 

No  concrete  shall  be  deposited  in  the  forms  when  the  temperature  drops 
below  20  degrees  Fahr.,  unless  such  forms  have  been  constructed  in  a  manner 
approved  by  the  Engineer,  to  prevent  freezing  of  the  concrete  mix. 

Joints. — Where  a  break  occurs  in  the  day's  pour,  no  additional  concrete 
shall  be  deposited  on  such  a  joint  when  work  is  subsequently  started  until 
the  joint  has  been  thoroughly  scrubbed  to  remove  all  laitance  and  other 
foreign  matter.  If  so  directed  a  layer  of  cement  grout  shall  be  deposited 
upon  the  joint  immediately  before  placing  fresh  concrete. 

It  is  the  intent  of  these  specifications  to  secure  a  section  of  wall  between 
expansion  joints  free  of  all  joints  as  above  and  the  contractor  shall  use 
plant  of  such  capacity  that  a  section  can  be  poured  complete  in  a  regular 
day's  operation.  When,  due  to  an  emergency,  such  a  construction  joint  is 
unavoidable,  the  Engineer,  or  his  duly  authorized  representative  shall  in- 
struct the  Contractor  as  to  what  details  of  construction  must  be  adopted 
to  obtain  the  full  efficiency  of  such  a  joint  and  to  prevent,  as  far  as  possible, 
any  unsightly  appearance  of  the  face  of  the  wall  after  the  forms  have  been 
stripped. 

Drains. — There  shall  be  incorporated  in  the  wall,  tile  drains  of  spacing 
and  diameter  shown  on  the  plans.  Immediately  back  of  the  drains  shall 
be  placed  one  cubic  yard  of  broken  stone. 

Waterproofing. — The  back  of  the  retaining  walls  shall  be  given  two  coats 
of  hot  asphalt  or  pitch.  The  back  of  the  wall,  before  the  tar  is  applied  shall 
be  thoroughly  dried  and  free  of  all  frost. 

(Insert  specifications  for  tar  as  given  on  page  240.) 

Extreme  care  shall  be  exercised  in  placing  the  fill  back  of  the  wall  so  that 
the  coats  of  tar  shall  not  be  abraded. 

If,  after  the  fill  has  been  in  place  the  face  of  the  wall  shows  evidence  of 
water  leaking  through  it,  the  contractor,  if  so  directed  by  the  Engineer, 
shall  excavate  back  of  the  wall  to  the  indicated  position  of  the  defective 


SPECIFICATIONS  257 

waterproofing  and  shall  make  such  repairs  as  are  necessary,  no  additional 
payment  to  be  made  for  this  work 

Concrete  Finish. — Where  no  special  face  finish  is  indicated,  the  Contractor 
shall,  immediately  upon  removing  the  forms,  remove  all  wires,  rods,  etc., 
or  cut  them  back  to  about  two  inches  from  the  face  of  the  wall.  He  shall 
then  point  up  these  places  with  a  rich  mortar  or  concrete.  The  face  of  the 
wall  will  then  be  rubbed  down  with  suitable  appliances  as  approved  by  the 
Engineer  and  the  entire  surface  given  a  coat  of  thin  grout  wash. 

Reinforcing  Bars. — Reinforcing  bars  shall  be  placed  in  the  concrete  walls 
of  dimensions  and  spacing  as  shown  on  the  plans.  Payment  for  these  rods 
includes  all  labor  and  material  required  for  their  installation  as  indicated. 

Rods  shall  be  deformed  as  approved  by  the  Engineer.  Plain  bars  may 
not  be  used. 

Rods  shall  be  bent  to  radii  as  indicated  and  shall  generally  be  delivered 
in  the  full  length  as  required  on  the  plans. 

Rods  shall  be  made  by  the  open  hearth  process  with  the  following  maxi- 
mum impurities: 

Phosphorus,  not  more  than  0.04  per  cent. 

Sulphur,  not  more  than  0.05  per  cent. 

The  elastic  limit  or  yield  point  shall  not  be  less  than  40,000  pounds  per 
square  inch. 

Test  specimens  for  bending  shall  be  bent  under  the  following  conditions 
without  fracture  on  the  outside  of  the  bent  portion: 

Around  twice  their  own  diameter. 
1  in.  in  diam.,  80  degrees. 
%  in.  in  diam.,  90  degrees. 
3^  in.  in  diam.,  110  degrees. 

Around  their  own  diameter. 

Y±  in.  in  diam.,  130  degrees. 

21  e  in-  in  diam.,  140  degrees. 

%  in.  or  less  in  diam.,  180  degrees 


Retaining  Walls,  Including  Lateral  Earth  Pressure1 

ALEXANDER,  T.,  and  THOMSON,  A.  W.  Elemental  Applied  Mechanics. 
575  p.  1902.  Contains  chapters,  "  Application  of  the  Ellipse  of  Stress 
to  the  Stability  of  Earthwork,"  p.  70-86,  and  "The  Scientific  Design 
of  Masonry  Retaining  Walls." 

ALLEN,  J.  ROMILLY.  Investigation  of  the  Question  of  the  Thrust  of  Earth 
Behind  a  Retaining  Wall.  3  diag.  1877.  (Van  Nostrand's  Engineer- 
ing Magazine,  v.  17,  p.  155-158.)  Mathematical  solution. 

ALLEN,  KENNETH.  Design  of  Retaining  Walls.  1892.  (Engineering  Rec- 
ord, v.  26,  p.  341-342,  356-357,  374,  393.)  On  practical  design  of 
retaining  walls,  sea  walls,  and  dock  walls.  Illustrated  with  actual 


AMERICAN  RAILWAY  ENGINEERING  AND  MAINTENANCE  OF  WAY  ASSOCIA- 
TION.    [Report  of  Committee  on]  Retaining  Walls  and  Abutments. 
1909.     (Proceedings,  Tenth  Annual  Convention,  Am.  Ry.  Eng.  and 
1  From  Report  Spec.  Comm.  on  Soils  A.S.C.E. 

17 


258  RETAINING  WALLS 

Maintenance  of  Way  Assoc.,  p.  1317-1337.)     Gives  information  show- 
ing practice  of  various  railroads  in  the  designing  of  retaining  walls. 
Committee  submits  method  of  determining  earth  pressures  based  on 
Rankine's  formula. 
— Condensed.     1909.     (Engineering  Record,  v.  60,  p.  288-290.) 

AUDE.  Nouvelles  Experiences  sur  la  Poussee  des  Terres.  1849.  (Comptes 
Rendus  Hebdomadaires  des  S6ances  de  1' Academic  des  Sciences,  v.  28, 
p.  565-566.)  Short  review  of  Audi's  work  presented  by  Poncelet. 

BAKER,  BENJAMIN.  Actual  Lateral  Pressure  of  Earthwork.  1881.  (Min- 
utes of  Proceedings,  Inst.  C.  E.,  v.  65,  p.  140-186.)  Discussion,  p.  187- 
241.  Aims  to  present  data  on  actual  lateral  pressure  of  earthwork,  as 
distinguished  from  "text-book"  pressures,  which  latter  the  author 
holds  to  be  generally  incorrect 

—1881.     (Van  Nostrand's  Engineering  Magazine,  v.  25,  p.  333-342,  353- 
371,  492-505.) 

BARD  WELL,  F.  W.  Note  on  the  "Horizontal  Thrust  of  Embankments." 
1861.  (Mathematical  Monthly,  v.  3,  p.  6-7.)  Finds  the  formula  de- 
rived by  D.  P.  Woodbury  to  be  correct. 

BOARDMAN,  H.  P.  Concerning  Retaining  Walls  and  Earth  Pressures. 
1905.  (Engineering  News,  v.  54,  p.  166-169.)  Concludes  that  in- 
formation regarding  earth  pressures  is  quite  inexact.  Suggests  con- 
ducting series  of  tests  on  large  scale. 

BONE,  EVAN  P.  Reinforced  Concrete  Retaining  Wall  Design.  1907. 
(Engineering  News,  v.  57,  p.  448-452.)  Calculations  of  earth  pressures, 
and  diagrams. 

BOUSSINESQ,  J.  Calcul  Approche  de  la  Poussee  et  de  la  Surface  de  Rupture, 
dans  un  Terre-plein  Horizontal  Homogene,  Contenu  par  un  Mur 
Vertical.  1884.  (Comptes  Rendus  Hebdomadaires  des  Seances  de 
1' Academic  des  Sciences,  v.  98,  p.  790-793.) 

BOUSSINESQ,  J.  Complement  a  de  Prece"dentes  Notes  sur  la  Poussee  des 
Terres.  1884.  (Annales  des  Fonts  et  Chausstes,  ser.  6,  v.  7,  p.  443-481.) 

BOUSSINESQ,  J.  Equilibrium  of  Pulverulent  Bodies.  1  diag.  1877. 
(Minutes  of  Proceedings,  Inst.  C.  E.,  v.  51,  p.  277-283.)  Abstract 
translation  of  "Essai  The"orique  sur  1'Equilibre  des  Massifs  Pulve"ru- 
lents,  Compare  a  celui  de  Massifs  Solides  et  sur  la  Pouss6e  des  Terres 
sans  Cohesion."  Brussels.  1876. 

1881.     (Van  Nostrand's  Engineering  Magazine,  v.  25,  p.  107-110.) 

BOUSSINESQ,  J.  Int6gration  de  1'Equation  Differentielle  qui  peut  Donner 
une  Deuxieme  Approximation,  dans  le  Calcul  Rationnel  de  la  Poussee 
Exercee  contre  un  Mur  par  des  Terres  Depourvues  de  Cohesion.  1 
diag.  1870.  (Comptes  Rendus  Hebdomadaires  des  Seances  de  1'Acad- 
emie  des  Sciences,  v.  70,  p.  751-754.) 

BOUSSINESQ,  J.  Note  sur  la  Determination  de  1'Epaisseur  Minimum  que 
doit  avoir  un  Mur  Vertical,  d'une  Hauteur  et  d'une  Densite  Donn6es, 
pour  Contenir  un  Massif  Terreux,  sans  Cohesion,  dont  la  Surface 
Superieure  est  Horizontale.  1  diag.  1882.  (Annales  des  Ponts  et 
Chaussees,  ser.  6,  v.  3,  p.  625-643.)  Application  of  the  theory  of 
earth-pressure,  as  developed  by  Rankine  and  Darwin,  to  design  of 
vertical  walls. 


SPECIFICATIONS  259 

BOUSSINESQ,  J.  Note  sur  la  Me"thode  de  M.  Macquorn-Rankine  pour  le 
Calcul  des  Pressions  Exercees  aux  Divers  Points  d'un  Massif  Pesant 
que  Limite,  du  Cote  Supe*rieur,  une  Surface  Cylindrique  a  Generatrices 
Horizontals,  et  qui  est  Indefini  de  Tous  les  Autres  Cotes.  1874. 
(Annales  des  Fonts  et  Chaussees,  ser.  5,  v.  8,  p.  169-187.)  Criticism  of 
Rankine's  theory  of  earth  pressure. 

BOUSSINESQ,  J.  Sur  la  Poussee  d'une  Masse  de  Sable,  a  Surface  Superieure 
Horizontale,  Contre  une  Paroi  Verticale  dans  le  Voisinage  de  Laquelle 
son  Angle  de  Frottement  Interieur  est  Suppose  Croitre  Legerement 
d'apres  une  Certaine  Loi.  1884.  (Comptes  Rendus  Hebdomadaires  des 
Seances  de  1' Academic  des  Sciences,  v.  98,  p.  720-723.) 

BOUSSINESQ,  J.  Sur  la  Poussee  d'une  Masse  de  Sable,  a  Surface  Superieure 
Horizontale,  Contre  une  Paroi  Verticale  ou  Inclinee.  1884.  (Comptes 
Rendu?  Hebdomadaires  des  Seances  de  l'Acade"mie  des  Sciences,  v. 
98,  p.  667-670.) 

BOUSSINESQ,  J.  Sur  le  Principe  du  Prisme  de  plus  grande  Poussee  Pose  par 
Coulomb  dans  la  Theorie  de  1'Equilibre  Limite  des  Terres.  1884. 
(Comptes  Rendus  Hebdomadaires  des  Seances  de  1' Academic  des  Sciences, 
v.  98,  p.  901-904,  975-978.)  Critical  review. 

BOUSSINESQ,  J.  Sur  les  Lois  de  la  Distribution  Plane  des  Pressions  a  1' In- 
terieur des  Corps  Isotropes  dans  TEtat  d'Equilibre  Limite.  1874. 
(Comptes  Rendus  Hebdomadaires  des  Seances  de  1' Academic  des  Sciences, 
v.  78,  p.  757-759.) 

BOVEY,  HENRY  T.  Theory  of  Structures  and  Strength  of  Materials,  ed.  3. 
835  p.  1900.  Includes  section  on  earthwork  and  retaining  walls. 

BURSTING  PRESSURE  OF  AN  EARTH  FILL.  1912.  (Engineering  News,  v.  68, 
p.  593-594.)  Editorial  discussing  the  causes  of  failure  of  a  retaining 
wall  in  St.  Louis. 

CAIN,  WILLIAM.  Cohesion  and  the  Plane  of  Rupture  in  Retaining  Wall 
Theory.  1  diag.  1912.  (Engineering  News,  v.  67,  p.  992.)  Letter 
to  editor  discussing  Hirschthal's  article  "  Some  Contradictory  Retaining 
Wall  Results,"  Engineering  News,  v.  67,  p.  799. 

CAIN,  WILLIAM.  Earth  Pressure,  Retaining  Walls  and  Bins.  287  p.  1916. 
Wiley.  Contains  chapters  on  the  theory  of  earth  friction  and  cohesion, 
of  earth  thrust,  and  of  'bins.  Gives  special  attention  to  coherent  and 
non-coherent  earths.  Emphasizes  throughout  the  presence  in  earth  of 
cohesion  as  well  as  of  friction. 

CAIN,  WILLIAM.  Retaining  Walls.  1880.  (Van  Nostrand's  Engineering 
Magazine,  v.  22,  p.  265-277.)  Considers  "the  earth  as  a  homogeneous 
and  incompressible  mass,  made  up  of  little  grains,  possessing  the  resis- 
tance to  sliding  over  each  other  called  friction,  but  without  cohesion." 

CALCULATIONS  FOR  RETAINING  WALLS.  1911.  (Architect  and  Contract  Re- 
porter, v.  86,  p.  43-44,  59-61,  75-76,  85-87,  96-97,  109-110.)  Takes 
all  factors  into  consideration,  wind  pressure,  slides,  earth  pressure,  etc. 
"Angles  of  repose  of  various  earths,"  p.  109. 

CARTER,  FRANK  H.  Bracing  and  Sheeting  Trenches.  1910.  (Engineering- 
Contracting,  v.  34,  p.  76-78.)  Computes  pressures  on  bracing  and  shor- 
ing for  well  under-drained  excavations  in  virgin  soil. 

CARTER,  FRANK  H.     Comparative  Sections  of  Thirty  Retaining  Walls,  and 


260  RETAINING  WALLS 

Some  Notes  on  Retaining  Wall  Design.     1910.     (Engineering  News,  v. 
64,  p.  106-108.)     Discusses  theoretical  earth  pressures,  giving  formulas. 

CLAVENAD.  Memoire  sur  la  Stabilite,  les  Mouvements,  la  Rupture  des 
Massifs  en  General,  Coherents  ou  sans  Cohesion.  Quelques  Consider- 
ations sur  la  Poussee  des  Terres,  Etude  Speciale  des  Murs  de  Soutene- 
ment  et  de  Barrages.  64diag.  1887.  (Annales  des  Fonts  et  Chaussees, 
ser.  6,  v.  13,  p.  593-683.) 

COLEMAN,  T.  E.  Retaining  Walls  in  Theory  and  Practice;  A  Text-book  for 
Students.  160  p.  1909.  Design  and  construction.  Avoids  advanced 
mathematics  where  possible. 

CONSIDERE.  Note  sur  la  Poussee  des  Terres.  1870.  (Annales  des  Fonts 
et  Chaussees,  ser.  4,  v.  19,  p.  547-594.)  Extension  of  Levy's  theory  of 
earth-pressure.  See  Comptes  Rendus  Hebdomadaires  des  Stances  de 
1'Academie  des  Sciences,  v.  68,  p.  1456. 

CONSTABLE,  CASIMIR.  Retaining  Walls:  An  Attempt  to  Reconcile  Theory 
with  Practice.  3  diag.  1874.  (Transactions,  Am.  Soc.  C.  E.,  v.  3, 
p.  67-75.)  Gives  results  of  a  number  of  experiments  with  models, 
using  walls  made  of  wood  blocks  and  filling  composed  of  oats  and  peas. 

Abstract.     1873.      (Van   Nostrand's   Engineering   Magazine,   v.    8,    p. 

375-377.) 
—Condensed.     1873.     (Journal,  Franklin  Inst.,  v.  95,  p.  317-322.) 

CORNISH,  L.  D.  Earth  Pressures :  A  Practical  Comparison  of  Theories  and 
Experiments.  1916.  (Transactions,  Am.  Soc.  C.  E.,  v.'  81,  p.  191-201.) 
Discussion,  p.  202-221.  Endeavors  to  show  graphically  the  results 
obtained  in  actual  wall  design  by  the  use  of  the  different  formulas 
(principally  those  of  Rankine  and  Cain)  and  by  values  obtained  in 
certain  experiments,  so  that  the  points  of  interest  may  be  discussed 
without  resorting  to  mathematics. 

CORNISH,  L.  D.  Fallacies  in  Retaining  Wall  Design  and  the  Lateral  Pres- 
sure of  Saturated  Earth.  1916.  (United  States  Corps  of  Engineers, 
Professional  Memoirs,  v.  8,  p.  161-172.)  Discussion,  p.  173-195. 
Treats  of  lateral  pressure  of  saturated  soils  in  connection  with  the  de- 
sign of  retaining  walls.  Presents  considerable  mathematical  data  on 
the  treatment  of  saturated  soil  in  such  design  work. 

COUPLET.  De  la  Poussee  des  Terres  Centre  leurs  Revestemens  et  la  Force 
des  Revestemens  qu'on  Leur  Doit  Opposer.  8  pi.  1726-1728.  (His- 
toire  de  P Academic  Royale  des  Sciences,  v.  28,  p.  106-164;  v.  29,  p. 
132-141  ;v.  30,  p.  113-138. 

COUSINERY.  Determination  Graphique  de  1'Epaisseur  des  Murs  de  Soutcne- 
ment.  1  pi.  1841.  (Annales  des  Fonts  et  Chaussees,  ser.  2,  v.  2,  p. 
167-184.)  Develops  a  method  of  graphical  determination  of  thickness 
of  retaining  walls.  Shows  how  to  apply  the  theory  of  earth  pressure 
in  connection  with  this  graphical  construction. 

CRAMER,  E.  Die  Gleitflache  des  Erddruck-prismas  und  der  Erddruck  gegen 
geneigte  Stiitzwande.  4  diag.  1879.  (ZeitschriftfurBauwesen,v.29, 
p.  521-526.) 

CRELLE.  Zur  Statik  unf ester  Korper.  An  dem  Beispiele  des  Drucks  der 
Erde  auf  Futtermauern.  1  pi.  1850.  (Abhandlungen  der  Konig- 


SPECIFICATIONS  261 

lichen  Akademie  der  Wissenschaften  zu  Berlin,  v.  34,  p.  61-97.)  To  be 
found  in  section  "  Mathematische  Abhandlungen." 

CUNO.  Die  Steinpackungen  und  Futtermauern  der  Rhein-Nahe-Eisenbahn. 
1861.  (Zeitschrift  fur  Bauwesen,  v.  11,  p.  613-626.) 

CURIE,  J.  Note  sur  la  Brochure  de  M.  Benjamin  Baker  Intitulee:  "The 
Actual  Lateral  Pressure  of  Earthwork."  9  diag.  1882.  (Annales  des 
Fonts  el  Chaussees,  ser.  6,  v.  3,  p.  558-592.)  Criticism  of  Baker's  paper 
in  Minutes  of  Proceedings,  Inst.  C.  E.,  v.  65,  p.  140. 

CURIE,  J.  Nouvelles  Experiences  Relatives  a  la  Theorie  de  la  Poussee  des 
Terres.  4  diag.  1873.  (Comptes  Rendus  Hebdomadaires  des  Seances 
de  P Academic  des  Sciences,  v.  77,  p.  142-146.) 

CURIE,  J.  Sur  la  Poussee  des  Terres  et  la  Stabilite  des  Murs  de  Revetments. 
1868.  (Comptes  Rendus  Hebdomadaires  des  Seances  de  1'Academie  des 
Sciences,  v.  67,  p.  1216-1218.)  Theoretical  paper. 

CURIE,  J.  Sur  la  Theorie  de  la  Poussee  des  Terres.  1871.  (Comptes 
Rendus  Hebdomadaires  des  Seances  de  1'Academie  des  Sciences,  v.  72, 
p.  366-369.)  Critical  review  of  the  theories  advanced  by  Maurice 
Levy. 

CURIE,  J.  Sur  la  Theorie  de  la  Poussee  des  Terres.  1  diag.  1873.  (Comp- 
tes Rendus  Hebdomadaires  des  Seances  de  1'Academie  des  Sciences,  v. 
77,  p.  778-781.)  Reply  to  Saint-Venant's  criticism  in  same  volume. 

CURIE,  J.  Sur  le  Disaccord  qui  Existe  entre  1'Ancienne  Theorie  de  la 
Poussee  des  Terres  et  1'Experience.  1  diag.  1873.  (Comptes  Rendus 
Hebdomadaires  des  Seances  de  1'Academie  des  Sciences,  v.  76,  p.  1579- 
1582.) 

CURIE,  J.  Trois  Notes  sur  la  Theorie  de  la  Poussee  des  Terres.  Disaccord 
entre  1'Ancienne  Theorie  et  1'Experience;  Nouvelles  Experiences;  Re- 
ponse  aux  Objections.  1873.  Gauthier-Villars.  Paris.  X$75.  (An- 
nales des  Fonts  et  Chaussees,  ser.  5,  v.  9,  p.  490.)  Short  review  of  Curie's 
pamphlet. 

DALY,  CESAR.  Mur  de  Soutenment  de  la  Terrasse  du  Chateau  de  Meudon, 
1  pi.  1859.  (Revue  Generale  de  I' Architecture  et  des  Travaux  Publics, 
v.  17,  p.  243.) 

DIAGRAM  FOR  OVERTURNING  MOMENTS  ON  RETAINING  WALLS  FOR  EARTH 
or  Water.  1907.  (Engineering  News,  v.  57,  p.  460.)  Diagram  was 
constructed  by  Charles  H.  Hoyt. 

DONATH,  AD.  Untersuchungen  iiber  den  Erddruck  auf  Stiitzwiinde  ange- 
stellt  mit  der  fur  die  Technische  Hochschule  in  Berlin  erbauten  Versuchs- 
vorrichtung.  1  pi.  1891.  (Zeitschrift  fur  Bauwesen,  v.  41,  p.  491-518.) 

Du  Bois,  A.  J.  Upon  a  New  Theory  of  the  Retaining  Wall.  14  diag. 
1879.  (Journal,  Franklin  Inst.,  v.  108,  p.  361-387.)  Gives  a  concise 
history  of  the  subject,  and  develops  in  detail  Weyrauch's  theory. 

DUNCAN,  LINDSAY.  Plumbing  a  Leaning  Retaining  Wall  and  Bridge  Abut- 
ment. 1906.  (Engineering  News,  v.  55,  p.  386.) 

DYRSSEN,  L.  Analytische  Bestimmung  der  Lage  der  Stiitzlinie  in  Futter- 
mauern. 11  diag.  1885.  (Zeitschrift  fur  Bauwesen,  v.  35,  p.  101-106.) 

DYRSSEN,  L.  Ermittlung  von  Futtermauerquerschnitten.  1  diag.  1886. 
(Zeitschrift  fur  Bauwesen,  v-  36,  p.  389-392.) 


262  RETAINING  WALLS 

DYRSSEN,  L.  Ermittlung  von  Futtermauerquerschnitten  mit  gebogener 
oder  gebrochener  vorderer  Begrenzungslinie.  3  diag.  1886.  (Zeit- 
schrift fur  Bauwesen,  v.  36,  p.  127-130.) 

EDDY,  HENRY  T.  New  Constructions  in  Graphical  Statistics.  1877.  (Van 
Nostrand's  Engineering  Magazine,  v.  17,  p.  1-10.)  Contains  section  on 
"Retaining  Walls  and  Abutments,"  p.  5-10. 

ENGESSER,  FR.  Geometrische  Erddruck-Theorie.  1880.  (Zeitschrift  fur 
Bauwesen,  v.  30,  p.  189-210.) 

EVEREST,  J.  H.  Treatise  on  Retaining  Wall  Design.  1911  (Canadian  Engi- 
neer, v.  21,  p.  192-193,  237,  264-265.)  Considers  earth  Pressure,  slope, 
weights  of  materials,  etc. 

FLAMANT,  A.  Formules  Simples  et  tres  Approchees  de  la  Poussee  des  Terres, 
pour  les  Besains  de  la  Pratique.  1884.  (Comptes  Rendus  Hebdoma- 
daires  des  Seances  de  F  Academic  des  Sciences,  v.  99,  p.  1151-1153.) 

FLAMANT,  A.  Note  sur  la  Poussee  des  Terres.  1  pi.  1872.  (Annales  des 
Fonts  et  Chaussees,  ser.  5,  v.  4,  p.  242-275.)  Expounds  Rankine's 
theory. 

FLAMANT,  A.  Note  sur  la  Poussee  des  Terres.  1882.  (Annales  des  Fonts 
et  Chaussees,  ser.  6,  v.  3,  p.  616-624.)  Mostly  a  review  of  Baker's  paper 
in  Minutes  of  Proceedings,  Inst.  C.  E.,  v.  65,  p.  140. 

FLAMANT,  A.  Resume  d' Articles  Publics  par  la  Societe  des  Ingenieurs 
Civils  de  Londres  sur  la  Poussee  des  Terres.  1883.  (Annales  des 
Fonts  et  Chausses,  ser.  6,  v.  6,  p.  477-532.)  Review  of  Darwin's,  Gau- 
dard's  and  Boussinesq's  papers  in  Minutes  of  Proceedings,  Inst.  C.  E., 
v.  71  and  72. 

FLAMANT,  A.  Tables  Numeriques  pour  le  Calcul  de  la  Poussee  des  Terres. 
2  diag.  1885.  (Annales  des  Fonts  et  Chaussees,  ser.  6,  v.  9,  p.  515-540.) 
Gives  many  tables  of  constants  for  the  relations  derived  by  Boussinesq 
and  based  on  the  experiments  of  Darwin  in  England  and  Gobin  in 
France. 

GLAUSER,  J.  Bestimmung  der  Starke  geneigter  Stiitz — und  Futtermauern 
mit  Riicksicht  auf  die  Incoharenz  ihrer  Masse.  1880.  (Zeitschrift  fur 
Bauwesen,  v.  30,  p.  63-72.) 

GOBIN,  A.  Determination  Precise  de  la  Stabilite  des  Murs  de  Soutenement 
et  de  la  Pousse*e  des  Terres.  71  diag.  1883.  (Annales  des  Fonts  et 
Chaussees,  ser.  6,  v.  6,  p.  98-231.)  Points  out  some  faults  in  Rankine's 
theory,  develops  his  own  theory,  and  gives  various  applications  and 
results  of  experiments. 

GODFREY,  EDWARD.  Design  of  Reinforced  Concrete  Retaining  Walls. 
1906.  (Engineering  News,  v.  56,  p.  402-403.)  Considers  lateral  pres- 
sure of  different  materials,  angles  of  repose,  and  necessary  calculations. 

GOODRICH,  ERNEST  P.  Lateral  Earth  Pressures  and  Related  Phenomena. 
44  diag.,  3  dr.,  1  ill.  1904.  (Transactions,  Am.  Soc.  C.  E.,  v.  53,  p. 
272-304.)  Discussion,  p.  305-321.  Experimentally  determines  ratio  of 
lateral  to  vertical  pressure.  Gives  series  of  conclusions.  See  also  edi- 
torial, "Lateral  Earth  Pressure,"  Engineering  Record,  v.  49,  p.  633-634. 
— Abstract.  1904.  (Minutes  of  Proceedings,  Inst.  C.  E.,  v.  158,  p.  450- 
451.) 


SPECIFICATIONS  263 

GOULD,  E.  SHERMAN.     Retaining  Walls.     13  diag.     1877.     (Van  Nost rand's 

Engineering  Magazine,  v.  16,  p.  11-17.)     Methods  of  design. 
GOULD,  E.  SHERMAN.     Retaining  Walls.     2  diag.     1883.     (Van  Nostrand's 
Engineering  Magazine,  v.  28,  p.  204-207.)     Gives  the  theory  of  J. 
Dubosque. 
GRAFF,  C.  F.     High  Reinforced  Concrete  Retaining  Wall  Construction  at 

Seattle,  Wash.     1905.     (Engineering  News,  v.  53,  p.  262-264.) 
HIRSCHTHAL,  M.     Some  Contradictory  Retaining  Wall  Results.     1  diag. 
1912.     (Engineering  News,  v.  67,  p.  799-800.)     Letter  to  editor  re- 
viewing some  accepted  formulas  of  earth  pressure  on  retaining  walls. 
See  also  Cain,  Engineering  News,  v.  67,  p.  992. 

HISELY.  Constructions  Diverses  pour  Determiner  la  Poussee  des  Terres  sur 
un  Mur  de  Soutenement.  1899.  (Annales  des  Fonts  et  Chaussees,  ser. 
7,  v.  17,  p.  99-120.)  Develops  a  general  graphical  solution  applicable 
to  a  load  of  any  character. 

HOSKING.     On  the  Introduction  of  Constructions  to  Retain  the  Sides  of 
Deep   Cuttings  in  Clays,   or   Other  Uncertain  Soils.     14  dr.     1844. 
(Minutes  of  Proceedings,  Inst.  C.  E.,  v.  3,  p.  355-372.) 
— Condensed.     1846.     (Journal,  Franklin  Inst.,  v.  41,  p.  73-79.) 

HOWE,  MALVERD  A.  Retaining- Walls  for  Earth,  Including  the  Theory  of 
Earth-pressure  as  Developed  from  the  Ellipse  of  Stress,  with  a  Short 
Treatise  on  Foundations,  Illustrated  with  Examples  from  Practice, 
ed.  4.  167  p.  1907. 

HUGHES,  THOMAS.  Description  of  the  Method  Employed  for  Draining  some 
Banks  of  Cuttings  on  the  London  and  Croydon,  and  London  and  Bir- 
mingham Railways;  and  a  Part  of  the  Retaining  Wall  of  the  Euston 
Incline,  London  and  Birmingham  Railway.  4  ill.  1845.  (Minutes  of 
Proceedings,  Inst.  C.  E.,  v.  4,  p.  78-86.) 

INTERNATIONAL  CORRESPONDENCE  SCHOOLS.  Railroad  Location,  Railroad 
Construction,  Track  Work,  Railroad  Structures.  [473  p.]  (Inter- 
national Library  of  Technology,  v.  34B.)  Includes  section  on  theory 
and  design  of  retaining  walls,  p.  899-912. 

JACOB,    ARTHUR.     On  Retaining  Walls.     27  diag.     1873.     (Van  Nostrand's 
Engineering  Magazine,  v.  9,  p.  194-204.)     Reprint,  with  a  few  emenda- 
tions, of  author's  original  essay  on  "Practical  Designing  of  Retaining 
Walls."     Takes  up  design.     Considerable  attention  to  earth  pressure. 
—1873.     (Building  News,  v.  25,  p.  421-422,  465-466,  478-479.) 

JACQUIER.  Note  sur  la  Determination  Graphique  de  la  Poussee  des  Terres. 
5  diag.  1882.  (Annales  des  Fonts  et  Chaussees,  ser.  6,  v.  3,  p.  463-472.) 
Bases  his  graphical  construction  on  Rankine's  theory,  as  developed  by 
Levy,  Considere,  and  others.  / 

KIRK,  P.  R.  Graphic  Methods  of  Determining  the  Pressure  of  Earth  on 
Retaining  Walls.  1899.  (Builder,  London,  v.  77,  p.  233-235.) 

KLEIN,  ALBERT.  Die  Form  der  Winkelstutzmauern  aus  Eisenbeton  mit 
Riicksicht  auf  Bodendruck  und  Reibung  in  der  Fundamentfuge.  1909. 
(Beton  und  Eisen,  v.  8,  p.  384-387.) 

KLEITZ.  Determination  de  la  Poussee  des  Terres  et  Etablissement  des 
Murs  de  Soutenement.  1884.  (Annales  des  Fonts  et  Chaussees,  ser.  2, 
v.  7,  p.  233-256.)  Theoretical  discussion. 


264  RETAINING  WALLS 

KLEMPBEER,  F.  Graphische  Bestimmung  des  Erddruckes  an  eine  ebene 
Wand  mit  Riicksicht  auf  die  Cohasion  des  Erdreiches.  1  pi.  1870. 
(Zeitschrift,  Oesterreichischen  Ingenieur-und  Architekten-Vereines,  v. 
31,  p.  116-120.) 

KRANTZ,  J.  B.  Study  on  Reservoir  Walls ;  Translated  from  the  French  by 
F.  A.  Mahan.  54  p.  1883. 

LACHER,  WALTER  S.  Retaining  Walls  on  Soft  Foundations.  1915.  (Jour- 
nal, Western  Soc.  of  Engrs.,  v.  20,  p.  232-265.)  Experiments  gave  the 
following  conclusions  as  to  types  of  walls  and  their  advantages:  (1) 
The  block  wall  is  economical,  and  may  be  constructed  in  several  stages, 
but  it  does  not  possess  as  great  a  potential  factor  of  safety  as  a  mono- 
lithic wall;  (2)  the  heavy  batter  mass  wall  is  economical,  but  is  open  to 
the  same  objections  as  the  block  wall;  (3)  the  cellular  wall  offers  great 
resistance  to  overturning  or  sliding,  but  occupies  considerable  space 
before  filling  and  may  thus  interfere  with  use  of  tracks;  (4)  the  mass 
wall  on  piles  gives  maximum  security,  but  is  expensive  and  may  give 
trouble  because  of  damage  to  adjacent  buildings  on  insecure  founda- 
tions. 

LAFONT,  DE.  Memoire  sur  la  Pouss6e  des  Terres  et  sur  les  Dimensions  a 
Donner,  Suivant  leurs  Profils,  aux  Murs  de  Soutenement  et  de  Reser- 
voirs d'Eau.  1  pi.  1866.  (Annales  des  Fonts  et  Chaussees,  ser.  4,  v.  12, 
380-462.)  Gives  in  tabulated  form  experiments  performed  and  con- 
stants arrived  at  by  Aud6,  Domergue,  and  Saint-Guilhem,  p.  397-400. 

LAFONT,  DE.  Note  sur  la  Repartition  des  Pressions  dans  les  Murs  de 
Soutenement  et  de  Reservoirs,  Nouvelles  Formules  pour  le  Calcul  de 
ces  Murs.  1868.  (Annales  des  Fonts  et  Chaussees,  ser.  4,  v.  15,  p. 
199-203.) 

LAGRENE,  H.  DE.  Note  sur  la  Poussee  des  Terres  Avec  ou  Sans  Surcharges. 
8  diag.,  2  dr.  1881.  (Annales  des  Fonts  et  Chaussees,  ser.  6,  v.  2,  p. 
441-471.)  Gives  calculations  for  earth  pressure  of  level  surfaces  on 
vertical  retaining  walls. 

—Abstract.     1882.     (Minutes  of  Proceedings,  Inst.  C.  E.,  v.  68,  p.  336- 
337.) 

LATERAL  EARTH  PRESSURE.  1904.  (Engineering  Record,  v.  49,  p.  633-634.) 
Editorial  comment  on  "Lateral  Earth  Pressure  and  Related  Phenom- 
ena," by  Ernest  P.  Goodrich. 

LETHIER  and  JOZAN.  Note  sur  la  Consolidation  des  Terrassements  du 
Chemin  de  Fer  de  Gien  a  Auxerre.  2  pi.  1888.  (Annales  des  Fonts 
et  Chaussees,  ser.  6,  v.  16,  p.  5-18.)  Consolidation  of  treacherous  slopes 
in  heavy  cuts  by  means  of  rubble  spurs  perpendicular  to  face  of  slopes. 

Abstract  translation.     1889.     (Minutes  of  Proceedings,  Inst.  C.  E.,  v. 

95,  p.  466-468.) 

L'EVEILLE.  De  TEmploi  des  Contre-forts.  1844.  (Annales  des  Fonts  et 
Chaussees,  ser.  2  ,  v.  7,  p.  208-232.)  Derives  formulas  for  proper 
design. 

LEVY,  MAURICE.  Essai  sur  une  The"orie  Rationnelle  de  FEquilibre  des 
Terres  Fratchement  Remuees  et  ses  Applications  au  Calcul  de  la  Stabil- 
ite  des  Murs  de  Soutenement.  1869.  (Comptes  Rendus  Hebdomadaires 
des  Seances  de  P Academic  des  Sciences,  v.  68,  p.  1456-1458.)  Develops 


SPECIFICATIONS  265 

a  theory  of  earth  pressure,  and  shows  its  application  in  design  of  retain- 
ing walls. 

LEYGUE.  Notice  sur  les  grands  Murs  de  Soutenement  de  la  Ligne  de 
Mazamet  a  Bedarieux.  2  pi.  1887.  (Annales  des  Fonts  et  Chaussees, 
ser.  6,  v.  13,  p.  98-114.)  Considerable  attention  is  given  to  design. 

MACONCHY,  G.  C.  Earth-pressures  on  Retaining  Walls.  1898.  (Engi- 
neering, v.  66,  p.  256-257,  484-485,  641-643.)  Gives  simple  method 
for  calculating  overturning  moments. 

MAIN,  J.  A.  Graphic  Determination  of  Pressures  on  Retaining  Walls. 
1912.  (The  Engineer,  London,  v.  113,  p.  220.) 

MEEM,  J.  C.     Bracing  of  Trenches  and  Tunnels,  with  Practical  Formulas 

for  Earth  Pressures.     2  diag.,  5  ill.,  13  dr.     1908.     (Transactions,  Am. 

-    Soc.  C.  E.,  v.  60,  p.  1-23.)     Discussion,  10  diag.,  5  ill.     54  dr.,  p.  24- 

100.     Develops  a  theory  of  earth  pressure,  and  basis  of  this  theory 

deduces  analytical  relations. 

Abstract.     1908.     (Minutes  of  Proceedings,    Inst.    C.   E.,   v.    171,   p. 

435-436.) 

—Abstract.  1  ill.,  3  dr.  1907.  (Engineering  Record,  v.  56,  p.  494-496.) 
See  also  editorial  "Sheet  Piling  and  Earth  Pressure,"  p.  528,  and  letter 
to  editor,  p.  608. 

MEERIMAN,  MANSFIELD.  Text-book  on  Retaining  Walls  and  Masonry 
Dams.  122  p,  1893. 

MOFFET,  J.  S.  D.  Mistaken  Ideas  with  Reference  to  the  Resultant  Force 
and  the  Maximum  Pressure  in  Retaining  Wall  Calculations.  1903. 
(Feilden's  Magazine,  v.  9,  p.  197-199.) 

MOHLER,  C.  K.  Tables  for  the  Determination  of  Earth  Pressures  on  Re- 
taining Walls.  1909.  (Engineering  News,  v.  62,  p.  588-589.) 

MULLER-BRESLAU,  HEINRICH.  Erddruck  auf  Stiitzmauern.  159  p.  1906. 
"Literatur,"  p.  158-159.  Contains  a  thorough  discussion  of  the  theory 
of  the  lateral  pressure  of  sand  and  loose  earth,  and  a  full  description  of 
the  author's  extensive  experiments. 

PEARL,  JAMES  WARREN.  Retaining  Walls;  Failures,  Theories  and  Safety 
Factors.  1914.  (Journal,  Western  Soc.  of  Engrs.,  v.  19,  p.  113-172.) 
Discusses  foundation  soil  of  retaining  walls,  and  calculates  design 
mathematically. 

PETTERSON,  HAROLD  A.  Design  of  Retaining  Walls.  1908.  (Engineering 
Record,  v.  57,  p.  757-759,  777-778.)  Diagrams  are  given.  See  also 
letter  by  C.  E.  Day,  Engineering  Record,  v.  58,  p.  56. 

PICHAULT,  S.  Calcul  des  Murs  de  Soutenement  des  Terres  en  Cas  de  Sur- 
charges Quelconques.  1899.  (Memoires  et  Compte  Rendu  des  Travaux 
de  la  Societe  des  Ingenieurs  Civils  de  France,  1899,  pt.  2,  p.  210-266, 
844-846.)  .  Bibliography,  p.  264-266.  Mathematical  treatment  of 
earth  pressures  on  retaining  walls. 

PONCELET.  Memoir  e  sur  la  Stabilite  des  Rev  elements  et  de  leurs  Fondations. 
1840.  (Comptes  Rendus  Hebdomadaires  des  Seances  de  1' Academic  des 
Sciences,  v.  11,  p.  134-140.)  Review  of  the  author's  270-page  essay 
published  in  Memorial  de  VOfficier  du  Genie,  No.  13.  Author  is  an  able 
supporter  of  Coulomb's  theorv. 


266  RETAINING  WALLS 

Abstract.  1840.  (Revue  Generate  de  I' Architecture  et  des  Travaux 

Publics,  v.  1,  p.  482-483.) 

PRELINI,  CHARLES.  Graphical  Determination  of  Earth  Slopes,  Retaining 
Walls  and  Dams.  129  p.  1908.  Elementary  treatment,  for  students 
rather  than  professional  engineers.  Graphical  methods  are  given  for 
solving  problems  concerning  the  slopes  of  earth  embankments,  the 
lateral  pressure  of  earth,  and  the  thickness  of  retaining  walls  and 
dams. 

PURVER,  GEORGE  M.  Design  of  Retaining  Walls,  Adapted  from  Georg 
Christoph  Mehrtens,  "  Vorlesungen  iiber  Static  der  Baukonstructionen 
und  Festigkeitslehre."  1910.  (Engineering-Contracting,  v.  34,  p.  388- 
395.)  Includes  "Tables  for  Allowable  Pressure,  Adopted  by  the  Public 
Service  Convention  [Commission?],  First  District,  State  of  New  York." 

RAMISCH.  Neue  Versuche  zur  Bestimmung  des  Erddrucks.  1910.  (Zeit- 
schrift,  Oesterreichischen  Ingenieur-  und  Architekten-Vereines,  v.  62, 
p.  233-240;  v.  63,  p.  323-425.)  Mathematical  calculations. 

REBHANN,  GEORG.  Theorie  des  Erddruckes  und  der  Futtermauern  mit 
besonderer  Rucksicht  auf  das  Bauwesen.  1871.  (Zeitschrift,  Oester- 
reichischen Ingenieur-und  Architekten-Vereines,  v.  23,  p.  211.)  Review, 
by  O.  Baldermann,  of  Rebhann's  book,  published  in  1870  in  Vienna  by 
Carl  Gerold's  Son. 

REISSNER,  H.  Theorie  des  Erddrucks.  1910.  (Enzyklopadie  der  Mathe- 
matischen  Wissenschaften,  v.  4,  pt.  4,  p.  386-417.)  "Literatur,"  p.  387. 

REPPERT,  CHARLES  M.  Recent  Retaining  Wall  Practice,  City  of  Pitts- 
burgh. 1910.  (Proceedings,  Engrs.  Soc.  of  Western  Pennsylvania, 
v.  26,  p.  316-354.)  Discussion,  p.  355-367.  Givesatte  ntion  to  calcu- 
lation of  earth  pressures  as  affecting  design. 

RESAL,  JEAN.  Poussee  des  Terres.  2  v.  1903-1910.  (Enzyklopadie  des 
Travaux  Publics.)  v.  1.  Stabilite  des  Murs  de  Soutenement.  v.  2. 
Theorie  des  Terres  Coherentes. — Applications. — Tables  Numeriques. 
Purely  theoretical  work  on  earth  pressures  as  affecting  the  design  of 
structures,  v.l  deals  entirely  with  soils  lacking  cohesion. 

REUTERDAHL,  ARVID.  From  the  Soil  Up:  A  New  Method  of  Designing. 
1914.  (Engineering-Contracting,  v.  42,  p.  581-585.)  Considers  espe- 
cially retaining  wall  design.  Advocates  starting  with  the  bearing  capac- 
ity of  the  soil,  and  working  from  that  basis. 

ROSE,  W.  H.  Formulas  for  the  Design  of  Gravity  Retaining  Walls.  1910. 
(Engineering-Contracting,  v.  34,  p.  115-117.)  From  Professional  Mem- 
oirs, Corps  of  Engineers,  U.  S.  Army. 

SAINT-VENANT,  DE.  Examen  d'un  Essai  de  Theorie  de  la  Poussee  des  Terres 
Centre  les  Murs  Destines  a  les  Soutenir.  1873.  (Comptes  Rendus 
Hebdomadaires  des  Seances  de  1' Academic  des  Sciences,  v.  73,  p.  234- 
241.)  Criticizes  Curie's  theory,  and  defends  the  so-called  rational 
theory  developed  by  Levy. 

SAINT-VENANT,  DE.  Poussee  des  Terres.  Comparaison  de  ses  Evaluations 
au  Moyen  de  la  Consideration  Rationnelle  de  FEquilibre-limite,  et  au 
Moyen  de  1'Emploi  du  Principe  dit  de  Moindre  Resistance,  de  Moseley. 
1870.  (Comptes  Rendus  Hebdomadaires  des  Stances  de  1'Academie  des 
Sciences,  v.  70,  p.  894-897.) 


SPECIFICATIONS  267 

SAINT-VENANT,  DE.  Rapport  sur  un  Memoire  de  M.  Maurice  Levy,  Pre- 
sente  le  3  Juin,  1867,  Reproduit  le  21  Juin,  1869,  et  Intitule:  Essai  sur 
une  Theorie  Rationnelle  d'Equilibre  des  Terres  Fraichements  Remuees, 
et  ses  Applications  au  Calcul  de  la  Stabilite  des  Murs  de  Soutenement. 
1870.  (Comptes  Rendus  Hebdomadaires  des  Seances  de  I'Acad&me  des 
Sciences,  v.  70,  p.  217-235.)  Report  of  a  committee,  giving  a  historical 
review  of  the  works  on  earth  pressure,  and  discussing  in  detail  Maurice 
Levy's  theory. 

SAINT-VENANT,  DE.  Recherche  d'une  Deuxieme  Approximation  dans  le 
Calcul  Rationnel  de  la  Poussee,  Exercee,  Centre  un  Mur  dont  la  Face 
Posterieure  a  une  Inclinaison  quelconque,  par  des  Terres  non  Cohe- 
rentes  dont  la  Surface  Sup6rieure  s'Eleve  en  un  Talus  Plan  quelconque 
a  Partir  du  Haut  de  Cette  Face  du  Mur.  1  diag.  1870.  (Comptes 
Rendus  Hebdomadaires  des  Seances  de  1'Academie  des  Sciences,  v.  70, 
p.  717-724.)  Based  on  Levy's  theory. 

SAINT-VENANT,  DE.  Sur  une  Determination  Rationnelle,  par  Approxima- 
tion, de  la  Poussee  qu'  Exercent  des  Terres  Depourvues  de  Cohesion, 
Centre  un  Mur  ayant  une  Inclinaison  quelconque.  3  diag.  1870. 
(Comptes  Rendus  Hebdomadaires  des  Seances  de  1'Academie  des  Sciences 
v.  70,  p.  229-235,  281-286.)  Development  of  Levy's  theory. 

SAINT-VENANT,  DE.  Sur  une  Evaluation,  ou  Exacte  ou  d'une  Tres  Grande 
Approximation,  de  la  Poussee  des  Terres  Sablonneuses  Contre  un  Mur 
Destine  a  les  Soutenir.  1884.  (Comptes  Rendus  Hebdomadaires  des 
Seances  de  T Academic  des  Sciences,  v.  98,  p.  850-852.)  Based  on 
Boussinesq's  works. 

SCHAFFER.  Erddruck  und  Stiitzwande.  1  diag.,  1  pi.  1878.  (Zeitschrift 
fur  Bauwesen,  v.  28,  p.  527-548.) 

SCHMITT,  EDIT ARD  .  Empirische  Formeln  zur  Bestimmung  der  Starke  der  Fut- 
termauern.  1871.  (Zeitschrift,  Oesterreichischen  Ingenieur-und  Archi- 
tekten-Vereines,  v.  23,  p.  336-338.)  Mathematical  calculations  on  the 
basis  of  Rebhann's  tables. 

SCHWEDLER,  J.  W.  [Unterschnittene  Futtermauern.]  1871.  (Zeitschrift 
fur  Bauwesen,  v.  21,  p.  280-282.)  Discussion  of  the  formula  derived 
by  Schwedler  at  a  meeting  of  the  Architekten-Verein  zu  Berlin. 

SERBER,   D.   C.     Stability  of  Sea  Walls.     15  diag.     1906.     (Engineering 

News,  v.  56,  p.  198-200.)     Gives  method  of  design. 
—Brief  abstract.     1906.     (Le  Genie  Civil,  v.  50,  p.  32.) 

SHEET-PILING  AND  EARTH  PRESSURE.  1907.  (Engineering  Record,  v.  56, 
p.  528.)  Refers  particularly  to  paper  on  "The  Bracing  of  Trenches 
and  Tunnels,"  by  J.  C.  Meem. 

SIEGLER.  Experiences  Nouvelles  sur  la  Poussee  du  Sable.  1887.  (Annales 
des  Fonts  et  Chaussees,  ser.  6,  13,  p.  488-505.)  Experimental  method 
for  studying  reactions  between  masses  of  earth  and  their  supporting 
walls.  Friction  dynamometer  was  used  to  determine  intensity  of 
pressure. 

Condensed  translation.     "New  Experiments  on  the  Thrust  of  Sand." 

1887.     (Scientific  American  Supplement,  v.  34,  p.  9724-9725.) 

SINGER,  MAX.  Fliessende  Hange.  1902.  (Zeitschrift,  Oesterreichischen 
Ingenieur-  und  Architekten-Vereines,  v.  54,  pt.  1,  p.  190-196.)  De- 


268  RETAINING  WALLS 

scribes  yielding  of  sides  of  railway  cutting  in  valley  of  the  Eger,  Austria, 
with  methods  used  for  retaining  embankment. 

SINKS,  F.  F.  Analysis  and  Design  of  a  Reinforced  Concrete  Retaining  Wall. 
1905.  (Engineering  News,  v.  53,  p.  8-9.) 

SINKS,  F.  F.  Design  for  Reinforced  Concrete  Retaining  Wall.  1904. 
(Railroad  Gazette,  v.  37,  p.  676-677.)  Letter. 

SKIBINSKI,  CARL.  Ueber  Stiitzmauerquerschnitte.  1898.  (Zeitschrift, 
Oesterreichischen  Ingenieur-  und  Architekten-Vereines,  v.  45,  p.  666- 
670.) 

SKIBINSKI,  KARL.  Theorie  des  Erddrucks  auf  Grund  der  neueren  Versuchen. 
1  diag.,  1  pi.  1885.  (Zeitschrift,  Oesterreichischen  Ingenieur-  und 
Archi-tekten-Vereines,  v.  37,  p.  65-77.)  Develops  his  own  theory  of 
earth  pressure  based  on  the  experimental  work  of  Forchheimer,  Gobin, 
and  Darwin.  Gives  a  graphical  construction  of  his  theory,  and  methods 
of  practical  application. 

SPILLNER,  E.  Stiitzmauern.  1904.  (Handbuch  der  Architektur,  cd.  3. 
v.  3,  pt.  6,  p.  182-197.)  "Literatur,"  p.  196. 

STRUKEL,  M.  Beitrag  zur  .Kenntniss  des  Erddruckes.  2  diag.,  4  dr.  1888. 
(Zeitschrift,  Oesterreichischen  Ingenieur-  und  Architekten-Vereines,  v. 
40,  p.  119-125.)  Critical  review  of  the  salient  points  of  the  earth 
pressure  theory  as  developed  by  Coulomb,  Rebhann,  and  others.  In 
support  of  his  own  views,  gives  results  of  some  experiments. 

SYLVESTER,  J.  J.  On  the  Pressure  of  Earth  on  Revetment  Walls.  1  diag. 
1860.  (London,  Edinburgh  and  Dublin  Philosophical  Magazine  and 
Journal  of  Science,  ser.  4,  v.  20,  p.  489-499.)  Criticism  of  theories  of 
Coulomb  and  Rankine. 

TATE,  JAMES  S.  Surcharged  and  Different  Forms  of  Retaining  Walls.  59 
p.  18V3.  VanNostrand.  Theoretical  calculations  for  retaining  walls. 
— 1873.  (Van  Nostrand's  Engineering  Magazine,  v.  9,  p.  481-494.) 

THORNTON,  WILLIAM  M.  Retaining  Walls.  7  diag.  1879.  (Van  Nos- 
trand's Engineering  Magazine,  v.  20,  p.  313-318.)  Concise  and  simpli- 
fied account  of  the  theory  of  earth  pressure  and  its  application  to  the 
design  of  retaining  walls. 

VAN  BUREN,  JOHN  D.,  JR.,  Quay  and  Other  Retaining  Walls.  6  diag.  1872. 
(Transactions,  Am.  Soc.  C.  E.,  v.  2,  p.  193-221.)  Establishes  practical 
formulas  for  the  dimensions  of  walls  of  various  shapes  and  under  various 
conditions.  Follows  Coulomb's  theory.  An  appendix  gives  a  number 
of  mathematical  relations. 

VEDEL,  P.  Theory  of  the  Actual  Earth  Pressure  and  Its  Application  to 
Four  Particular  Cases.  1894.  (Journal,  Franklin  Inst.,  v.  138,  p. 
139-148,  189-198.)  Mathematical  calculation. 

WALMISLEY,  A.  T.  Retaining  Walls.  1907.  (The  Builder,  London,  v.  93, 
p.  647-648.)  Discusses  calculations  of  earth  pressure,  foundations,  etc. 

WEINGARTEN.  [Die  Theorie  des  Erddrucks.]  1  diag.  1870.  (Zeitschrift 
fur  Bauwesen,  v.  20,  p.  122-124.)  Abstract  of  a  paper  read  before  the 
Architekten-Verein  zu  Berlin. 

WESTON,  W.  E.  Tables  for  Use  in  Determining  Earth  Pressure  on  Retain- 
ing Walls.  1911.  (Engineering  News,  v.  65,  p.  756-757.) 

WINKLER,    E.     Neue   Theorie   des   Erddruckes.     19   diag.     1871.     (Zeit- 


SPECIFICATIONS  269 

schrift,  Oesterreichischen  Ingenieur-  und  Architekten-Vereines,  v.  23, 

p.  79-89,  117-122.) 
WOODBURY,  D.  P.     On  the  Horizontal  Thrust  of  Embankments.     1  diag. 

1859.     (Mathematical  Monthly,  v.  1,  p.  175-177.)     Mathematical  paper. 
WOODBURY,  D.  P.     Remarks  on  Barlow's  Investigation  of  "the  Pressure  of 

Banks,  and  Dimensions  of  Revetments."     2  diag.     1845.     (Journal, 

Franklin  Inst.,  v.  40,  p.  1-7.) 


o 


INDEX 


Numbers  refer  to  pages 


Abrams,   D.  A.,   concrete  strength, 

201,  216 
Abutments,  general  theory,  128 

highway,  132 

problems,  140 

settlement  cracks,  156 

types  of,  130 

Adhesion,  reinforced  concrete,  89 
Aggregates,      effect      on      concrete 
strength,  200 

fineness  modulus,  202,  216 

heating,  211 

proportions,  214 

ratio,  fine  to  coarse,  226 

surface  area,  202,  219 
Architectural  treatment,  232 
Arm,  vertical,  91 
Asphalt,  waterproofing,  240 


B 


Baker,  Sir  Benjamin,  3,  18 

Bars,  see  Rods. 

Bearing,  concrete  stress,  90 

Belidor,  2 

Bell,  cohesion,  23 

Bernoulli,  theory  of  flexure,  85 

Bibliography,  41,   77,   120. 

Special  Committee  on  Soil,  Am. 

Soc.  C.  E.,  257 
Bilger,   H.   E.,   standard  abutment 

sections,  132 
Board  marks,  232 
Bond,  see  Adhesion. 
Boussinesq,  J.,  2,  8,  31 
Box  sections,  132 

problem,  143 
Bracing,  188 


Bullet,  1 

Bureau    of    Standards,    Report    on 
Concrete,  199 


Cableway,  175 

Cain,  Wm.,  cohesion,  20,  22 

experimental  data,  19 

factor  of  safety,  57 

footing  of  counterfort  wall,  99 

modification    of    coulomb    the- 
ory, 5 

revetment  wall,  65 

surcharge,  28 

Calcium    chloride,    hardening    con- 
crete, 213 

Cement,  effect  on  concrete  strength, 
200 

Portland,  214 

proportions,  214 

specifications,  215 
Center  of  gravity,  walls,  63 

trapezoid,  10 
Clay,  as  a  foundation,  50 

failures,  162 

permissible  bearing,  52 
Codes,  building,  52 
Cofferdam,  pressures  on,  31 
Cohesion,  3,  20 
Colors,  face  treatment,  235 
Concrete,  acceleration  of  set,  212 

allowable  stresses,  90 

compressive  strength,  200 

construction,  197 

Cyclopean,  210 

distributing,  209 

materials,  213 

methods,  208 

methods  of  proportioning,  201 


271 


272 


INDEX 


Concrete,  pressures,  181,  183 

proportions,  214 

Report  Special  Committee,  Am. 
Soc.  C.  E.,  85 

Report  Tests  Bureau  of  Stand- 
ards, 199 

see    also    Aggregates,    Cement, 
Reinforced  concrete. 

trains,  173 

water  content,  197,  203 
Concreting,  winter,  210 
Conjugate  pressures,  7 
Coping,  rubble  walls,  229 
Cost  data,  248 

labor,  249 

rubble  walls,  231 
Counterfort,  design  of,  101 

economical  spacing,  150 
Counterfort  walls,  96,  107 

economic  comparison  with  "T" 

walls,  147 
Coulomb,  2,  5,  11 
Couplet,  2 

Crane,   erecting,   effect  upon  abut- 
ment, 129 
Cribbing,  concrete,  124 

timber,  124 
Crum,  R.  W.,  226 
Curves,  permissible  flattening,  243 


Details,  wall,  138 

Distributing  systems,  cableway,  170 

concrete,  170 

pneumatic,  170 

spouting,  170 

tower,  170,  171,  172 
Drainage,  238 

E 

Earth  pressure,  history  of  theory,  1 
problems,  36 
theories,  5 

Eddy,  Prof.,  theory  of  plates,  108 
Edwards,  L.  N.,  method  of  surface 

area,  219 

Embankment,     bounded     by     two 
walls,  126 


Embankment,  rolled  in  layers,  5 
see  also  Fill. 

Empiric  design,  3 

Enger,  M.  L.,  experiments  on  trans- 
mitted pressure,  31 

Equilibrium    polygon,    use    in   wall 
design,  48 

Error,   permissible  in  wall  survey, 
242 

Euler,  theory  of  flexure,  85 

Experimental  data,  18 


Face  treatment,  232 
Factor  of  safety,  48,  56,  84 
Failures,  wall,  57,  160 
Fill,  ideal  and  actual,  4 

sea  walls,  35 

Fineness  modulus,  202,  216 
Finish,  see  Face  treatment,   Archi- 
tectural treatment. 
Footing,  counterfort  wall,  98 

design  of  reinforced  concrete,  93 
Forms,  181,  187 

blaw,  191 

hydraulic  pressed  steel,  190 

lines  and  grades,  244 

oiling,  189 

on  curves,  194 

patent,  189 

problem  in,  195 

reuse,  187 

stripping,  188 

traveling,  193 
Foundation,  character  of,  49 

problems,  67 

see  also,  Rock,  Sand,  Clay,  Piles. 
Frame,  stresses  in  rigid,  132 
Friction,  between  wall  and  earth,  3, 
8,  19 

between  wall  and  foundation,  44 


G 


Grades,  242 

Gravel,  see  also  Aggregates. 

soil,  50,  52 

specifications  for  concrete,  216 


INDEX 


273 


Gravity  wall,  center  of  gravity,  63 

direct  design,  61 

merits,  137 

problems,  67 

stresses,  48 

table  of  dimensions,  64 

types,  65 
Godfrey,  E.,  97 

Goodrich,    E.     P.,    earth    pressure 
tests,  18 


II 


Hand  rail,  237 

Hell  Gate  Arch,  see  New  York  Con- 
necting Railroad.  • 

Hool,  Prof.,  factor  of  safety,  57 

Howe,  Prof.,  8,  23 

Husted,  A.  G.,  pressure  of  saturated 
soils,  32 


Interboro  Rapid  Transit  Co.,  East- 
ern Parkway  Walls,  127 
White  Plains  Road  Extension, 

127 
Isometric  drawing,  216 


Johnson,  N.  C.,  197 

Joints,  construction,  159,  233 

details  of,  158 

efficiency  of,  159 

expansion,  157 

omission  of,  158 
Joists,  forms,  186 

K 

Kelly,  E.  F.,  131 
Keys,  concrete,  209 


Labor,  costs,  249 
Lacher,  cellular,  123 

transmission  of  live-load,  30 

18 


Lagging,  forms,  184 
Levy,  M.,  2 
Lines  and  grades,  242 
Loads,  see  Pressure,  Surcharge. 
Love,  A.  E.  H.,  transmitted  pressure 
through  solids,  31 


M 


Mayniel,  2 

Mehrtens,  see  Purver,  G.  M. 

Middle  third,  56 

Mixer,  concrete,  see  Plant. 

Mixing,  proper  methods,  207 

time  of,  208 
Mohler,  C.  K.,  thrust  expression,  17 

wing-wall,  131 
Moments,  overturning,  44 

resistance    of     reinforced    con- 
crete, 87 

thrust  and  stability,  47 
Mortar,  rubble  wall,  227 


N 


Navier,  2 

Neutral  axis,  reinforced  concrete,  86 
New  York  Connecting  Railroad,  re- 
taining walls  of,  21,  127 


0 


Offset,  gravity  wall,  58 
Overturning,  criterion  against,  44 


Passive  stress,  23 
Piles,  50 

problems,  69 

proper  centering,  52 

walls  on,  77 
Plant,  165,  179 

arrangement,  166 

central,  168 

concrete,  see  Preface. 

rubble  walls,  227 

standard  layout,  166 
Plaster  coat,  231 


274 


INDEX 


Plates,  theory  of,  108 
Pointing,  stone  walls,  230 
Poncelet,  2,  5 

graphic    thrust    determination, 

38 

Pressure,  base,  distribution,  50 
cofferdam,  31 
permissible  soil,  52 
toe,  criterion  against  excessive, 

44 

values  of,  51 
see  also  Earth  pressure, 
transmission  of  vertical,  30 
Prior,  J.  H.,  abutments,  131 

cellular  walls,  123 
Public     Service     Commission,     see 

Codes,  Building. 
Purver,  G.  M.,  26 


R 


Rangers,  187 
Rankine,  2,  3,  5 

Reinforced  concrete,  abutments,  142 
constants,  88 
theory,  85 
walls,  79 

base  pressure,  83 
base  ratio,  82 
economical  width,  82 
factor  of  safety,  84 
merits,  137 
problems,  104 
skeleton  outline,  80 
tables,  84 

Reinforcement,  economical,  187 
see  also  Rods, 
shrinkage,  155 
supports,  192 
temperature,  154 
Resal,  2 

Robinson,  concrete  pressure  experi- 
ments, 182 
Rock,  49,  52 
Rods,  anchoring,  90 
bending,  90 
counterfort  walls,  101 
periphery  for  adhesion,  89 
see  also  Reinforcement, 


Rods,  specifications,  257 
Rondelet,  2 
Rubbing,  223 
Rubble,  cement,  46 

dry,  46 

Hetch-Hetchy  Railroad,  47 


Sallonmeyer,  2 
Sand,  foundation,  82 

see  also  Aggregates. 

specification,  215 
Serber,  D.  C.,  sea  walls,  35 
Settlement,  155 
Shale,  50 

Shear,  reinforced  concrete,  89 
Shrinkage,  155 

Shunk,    concrete    pressure     experi- 
ments, 182 
Slabs,  face,  counterfort  walls,  97 

see  Reinforced  concrete. 

thin,  139 

Sliding,  see  Friction. 
Soils,  bearing,  52 

plastic,  50 

saturated,  32 

see  also  Earth  pressure,  Fill, 
Foundations,  Pressures, 
etc. 

Specifications,  54 
Speedway,  cellular  walls  supporting, 

124 
Stone,  broken,  216 

see  Aggregates. 

St.  Venant,  theory  of  flexure,  85 
Sub-surface  structures,  136 
Surcharge,  25 

sea  walls,  35 
Surface  area,  202,  219 
Surveying,  242 
Sweeny,  F.  R.;  see  Cofferdam. 


Talbot,  Prof.  A.  N.,  197 
Tar,  specifications,  240 
Temperature,   distribution   in   large 

masses,  152 
stresses,  151 


INDEX 


275 


Thrust,  coulomb  expression,  14 
fluid  expression,  17 
Mohler,  C.  K.,  17 
Rankine  expression,  8 
standard  form,  9,  10,  15 

Tie-rods,  186 

Timber,  safe  stresses,  185 

Toe,  offset  gravity  walls,  59 

reinforced  concrete  walls,  95 

Tooling,  234 

Tower,  concrete,  176 

Track  elevation,  42,  123 

Trapezoid,  center  of  gravity,  8 

Trautwine,  3 


Vauban,  general,  1 

Volumes,  computation  of,  245 

W 

Walls,  Ashlar,  230 
backstays,  125 
cellular,  122 
classes,  43 
counterfort,  96 


Walls,  economic  location  and  height, 

42 

economy  of  various  types,  137 
European  practice,  126 
hollow  cellular,  123 
land  ties,  125 
relieving  arches,  125 
revetment,  65 
rubble  coping,  229 
construction,  228 
cost,  231 
face  finish,  230 
plant,  227 
specifications,  229 
sea,  34 

see  also,   Gravity  walls,    Rein- 
forced concrete  walls, 
selection    of    economical    type, 

147 

stone,  45 

Washers,  rod  anchorage,  91 
Water  content,  see  Concrete. 
Wedge  beam,  method  of  Cain,  97 
Wedge  of  maximum  sliding,  1 1 
Winter  concreting,  210 
Work,  theory  of  least,  132 


UNIVERSITY  OF  CALIFORNIA  LIBRARY 
This  book  is  DUE  on  the  last  date  stamped  below. 

Fine  schedule:  25  cents  on  first  day  overdue 

50  cents  on  fourth  day  overdue 
One  dollar  on  seventh  day  overdue. 


i^&m 

DEC  1 0  1947 
DEC  2  4  1947 
JAN  1  o  ]948 
MAY  10  1348 


lflji;»> 

NOV  s 

DEC 

fee  2 

MAY  21  1949 
1UN     ?  19^ 

i/wl»  /  I 

JUN  2     19521 
NOV141949kVAN  22  '*3F 


LIBRARY 

JAM  1 1  1950 


LD  21-100m-12,'46(A2012sl6)4l20 


DEC  28  19fel 


MAY  26 


195^ 


UNIVERSITY  OF  CALIFORNIA 

APARTMENT  OF  CIVIL   ENGINEERING 

BERKELEY,  CALIFORNIA 


Tfmo 

X 


Engineering 
Library 

UNIVERSITY  OF  CALIFORNIA  LIBRARY 


